On each tile, two quarter-circles connect midpoints of adjacent sides.
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Midpoints were used in cases where company disclosed a forecast range.
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Consider the letter X: two symmetrical diagonal lines, crossing at their midpoints, right?
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Women tend to have earlier midpoints than men, he noted, a difference of up to two hours.
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Once they gained this access, they were able to observe the adversary in action from these midpoints.
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Herbalife – Herbalife raised its earnings outlook for the year, but lowered the midpoints of its revenue outlook.
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It's a play about the failure of justice when it deals with unknowable midpoints instead of obvious extremes.
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Tall, long-leaf pines snapped like pencils at their midpoints, their downturned crowns often all pointing in the same direction.
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Unlike previous episodes of yuan weakness, the PBOC has only set firmer-than-expected official midpoints and made rounds of verbal comments to guard its currency this time.
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Unlike previous episodes of yuan weakness, the PBOC has recently only set firmer-than-expected official midpoints and made rounds of verbal comments to guard its currency this time.
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The midpoints of those ranges are below analysts' expectations of a profit of 50 cents per share and revenue of $389.5 million, according to Thomson Reuters I/B/E/S.
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SHANGHAI (Reuters) - China's yuan edged up against the dollar on Friday and is set for a mild weekly gain after the central bank fixed its daily midpoints stronger over the past week.
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Each contain an oval in the center, and at eight points along the circumference of each oval, beams of light stream toward the four corners of the sheet and the midpoints of its borders.
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Amid worries of a longer and costlier trade war, China has managed to keep the yuan confined to a relatively steady and narrow range for weeks by persistently setting midpoints that were stronger than markets expected.
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According to Sarah Binder of George Washington University, in the mid-20th century the voting records of over 30% of federal legislators were closer to the overall centre than they were to the midpoints of those representatives' political parties.
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"Collins expects both of these oscillators to cross above their midpoints in the near future, which will serve as a trigger telling all the other chart watchers out there that it is time to buy the stock of Qualcomm," Cramer said.
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SHANGHAI (Reuters) - China tweaked its formula for setting daily reference midpoints for its yuan currency on Monday, three sources with direct knowledge of the matter said, in what was seen as authorities' latest move to help curb speculation in the currency.
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After guiding the yuan sharply lower and pushing onshore spot rates to a five-year low earlier this month, the People's Bank of China (PBOC) has helped the currency steady by setting a succession of midpoints within a tight range.
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The People's Bank of China has set stronger than expected yuan daily midpoints for onshore trade and state banks were seen offering dollars in the past few days, in order to stop the yuan from falling too quickly and to stabilise market expectations.
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While places such as Qatar and Abu Dhabi compete for those flying between Europe, Asia and America, who can choose to make stopovers in any number of midpoints, American cities do not really benefit from such international layover traffic—or feel the need to compete for it.
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The supportive yuan midpoints and surge in borrowing rates suggested the People's Bank of China might have stepped up its action to defend the yuan to keep it from breaching the 7 per dollar level, Ken Cheung, Asian FX strategist at Mizuho Bank in Hong Kong, said in a note to clients.
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"Compression Line" had its first incarnation in 1968, at El Mirage Dry Lake Bed, in the Mojave: an open-topped sixteen-foot-long plywood trapezoid that Heizer and Hank Lee buried in a pit and, using shovels, packed with dirt on either side, until the pressure forced the midpoints to cave in, leaving two triangular voids kissing at their tips.
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The secondary lattice tracks the midpoints of the bonds in the system, and forbids the overlap of bond midpoints. This effectively leads to disallowing polymers from crossing each other.
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Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords. A circle has the following property: : The midpoints of parallel chords lie on a diameter. An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.) ; Definition: Two diameters d_1,\, d_2 of an ellipse are conjugate if the midpoints of chords parallel to d_1 lie on d_2\ .
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The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).
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The Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral that is not a parallelogram. The line segments connecting the midpoints of opposite sides of a convex quadrilateral intersect in a point that lies on the Newton line.
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All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints of their edges.
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Royal Soc. Edinburgh, 136A, 1195−1205. In the figure the red curve is the curve γ, the black lines are the tangent line and some near-by tangent lines, the black dots are the midpoints on the displayed lines, and the blue line is the locus of midpoints.
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Midpoints were known and used to calculate Arabian Lots or Parts, like part of fortune in the 3rd century. Guido Bonati used direct midpoints (1123-1300) in the 13th century to refine timings in an event chart. Alfred Witte was the first person to do a lot of investigation on midpoints using movable dials and together with Ludwig Rudolph and Herman Lefeldt formed the Hamburg School of Astrology and the technique with the use of Trans Neptunian points was called the Uranian Astrology.Alfred Witte & Friedrich Sieggrün: Immerwährende Ephemeride, Witte-Verlag, Hamburg 1935.
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The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.
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The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
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The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
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The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are concurrent at a point called the "vertex centroid" and are all bisected by this point.Altshiller-Court, Nathan, College Geometry, Dover Publ., 2007.
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All the centers of inellipses of a given quadrilateral fall on the line segment connecting the midpoints of the diagonals of the quadrilateral.
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The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are concurrent at (all intersect at)a point called the "vertex centroid", which is the midpoint of all three of these segments.Altshiller- Court, Nathan, College Geometry, Dover Publ., 2007.
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Midpoints of parallel chords A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry (see Axis-direction of a parabola).
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Points J, K, and L are the midpoints of the line segments between each altitude's vertex intersection (points A, B, and C) and the triangle's orthocenter (point S). For an acute triangle, six of the points (the midpoints and altitude feet) lie on the triangle itself; for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.
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On each parallel line we mark the midpoint of the line segment joining these two intersection points. For each parallel line we get a midpoint, and so the locus of midpoints traces out a curve starting at p. The limiting tangent line to the locus of midpoints as we approach p is exactly the affine normal line, i.e. the line containing the affine normal vector to γ(I) at γ(t0).
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Its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either.
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The Roman calendar, in turn, had Celtic origins. Candlemas concurs with Imbolc, one of the Celtic 'cross- quarter days', the four days which marked the midpoints between solstice and equinox.
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A smaller square (tilted 45 degrees) is made from the midpoints of the large square. Connecting the midpoints of the small square and extending the lines to the edge of the circle will form the arms of the cross, otherwise known as the "first" and "last" steps of the chakana. Lines are drawn from the points the lines exit the circle, to complete the cross. A small circle is made from the diameter of the cross lines.
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For s=t=\tfrac 1 2 the points of contact U,V,W are the midpoints of the sides and the inellipse is the Steiner inellipse (its center is the triangle's centroid).
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The rectified 24-cell can be derived from the 24-cell by the process of rectification: the 24-cell is truncated at the midpoints. The vertices become cubes, while the octahedra become cuboctahedra.
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Bonatti's dates of birth and death are unknown, the latter probably occurring between 1296 and 1300. In 1233 he is known as the winner of a dispute in Bologna with the friar Giovanni Schio from Vicenza, who maintained the non-scientific basis of astrology. He is probably the first astrologer to have used the midpoints in astrology. He used it to refine the timing for the military campaigns for the Count of MontefeltroMichael Harding and Charles Harvey: Working with Astrology: The Psychology of Harmonics, Midpoints and AstroCartoGraphy, Penguin Group, London.
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The line segments GH and IJ that connect the midpoints of opposite sides (the bimedians) of a convex quadrilateral intersect in a point that lies on the Newton line. This point K bisects the line segment EF that connects the diagonal midpoints. By Anne's theorem and its converse, any interior point P on the Newton line of a quadrilateral ABCD has the property that :[ABP]+[CDP]=[ADP]+[BCP], where [ABP] denotes the area of triangle ABP. If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line.
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P lies on the Newton line EF In Euclidean geometry Newton's theorem states that in every tangential quadrilateral other than a rhombus, the center of the incircle lies on the Newton line. Let ABCD be a tangential quadrilateral with at most one pair of parallel sides. Furthermore, let E and F the midpoints of its diagonals AC and BD and P be the center of its incircle. Given such a configuration the point P is located on the Newton line, that is line EF connecting the midpoints of the diagonals.
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Equilateral triangles on the sides of an arbitrary hexagon If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle.
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In addition to the sides and diagonals of a quadrilateral, some important segments are the two bimedians (connecting the midpoints of opposite sides) and the four maltitudes (each perpendicularly connecting one side to the midpoint of the opposite side).
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The Droz-Farny line theorem says that the midpoints of the three segments A_1A_2, B_1B_2, and C_1C_2 are collinear. The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof.
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A cyclic polygon (green), its midpoint polygon (red), and its midpoint- stretching polygon (pink) In geometry, the midpoint-stretching polygon of a cyclic polygon is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of .. It may be derived from the midpoint polygon of (the polygon whose vertices are the edge midpoints) by placing the polygon in such a way that the circle's center coincides with the origin, and stretching or normalizing the vector representing each vertex of the midpoint polygon to make it have unit length.
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File:Nine-point circle.svg The diagram above shows the nine significant points of the nine-point circle. Points D, E, and F are the midpoints of the three sides of the triangle. Points G, H, and I are the feet of the altitudes of the triangle.
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The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. If the quadrilateral is convex or concave (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.
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Although he is credited for its discovery, Karl Wilhelm Feuerbach did not entirely discover the nine-point circle, but rather the six point circle, recognizing the significance of the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle.
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PG(3,2) can be represented as a tetrahedron. The 15 points correspond to the 4 vertices + 6 edge-midpoints + 4 face-centers + 1 body-center. The 35 lines correspond to the 6 edges + 12 face-medians + 4 face-incircles + 4 altitudes from a face to the opposite vertex + 3 lines connecting the midpoints of opposite edges + 6 ellipses connecting each edge midpoint with its two non-neighboring face centers. The 15 planes consist of the 4 faces + the 6 "medial" planes connecting each edge to the midpoint of the opposite edge + 4 "cones" connecting each vertex to the incircle of the opposite face + one "sphere" with the 6 edge centers and the body center.
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In this case the vertices of A1 are the midpoints of the sides of the quadrilateral A0 and those of A2 are the apices of the triangles with apex angles π/2 erected over the sides of A1. The PDN-theorem asserts that A2 is a square in this case also.
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Annals of the Brentwood (N.H.) Congregational Church and Parish Residents petitioned for a dividing line between the midpoints of the northern and southern boundaries. In 1744, Gov. Wentworth issued a King's Patent to establish a new town called Keeneborough Parish, named after his friend, Sir Benjamin Keene (1697–1757), English minister to Spain.
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Ebertin utilized the research on astrological midpoints of Hamburg School surveyor and astrologer Alfred Witte first published in 1928 in Witte's Regelwerk für Planetenbilder. Shortly after Witte's death in 1941, Ebertin used Witte's extensive research on astrological midpoints, and a 4th-harmonic "90° dial" developed by the Hamburg School of Astrology as the foundations of his School of Cosmobiology. Ebertin continued to promote astrological research, including medical applications of astrology while non-compliant Hamburg School astrologers were interned by the Third Reich, their books and publications banned. Reinhold Ebertin's main reference text on Cosmobiology entitled The Combination of Stellar Influences, sometimes referred to as the 'CSI' or the 'COSI', was inspired by Alfred Witte's Rulebook of Planetary Pictures [Ebertin 1972, p.
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In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus : a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2 where x is the distance between the midpoints of the diagonals. This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law. The German mathematician Carl Anton Bretschneider derived in 1842 the following generalization of Ptolemy's theorem, regarding the product of the diagonals in a convex quadrilateralAndreescu, Titu & Andrica, Dorian, Complex Numbers from A to...Z, Birkhäuser, 2006, pp. 207–209.
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For Yaglom, a hyperbola is a Minkowskian circle as in the Minkowski plane. Yaglom's description of this geometry is found in the "Conclusion" chapter of a book that initially addresses Galilean geometry.Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, page 193 He considers a triangle inscribed in a "circumcircle" which is in fact a hyperbola. In the Minkowski plane the nine-point hyperbola is also described as a circle: :...the midpoints of the sides of a triangle ABC and the feet of its altitudes (as well as the midpoints of the segments joining the orthocenter of △ABC to its vertices) lie on a [Minkowskian] circle S whose radius is half the radius of the circumcircle of the triangle.
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For the 2005 season, Guintoli returned to the Équipe du France GP Scrab, partnered first by Grégory Leblanc and then by Mathieu Gines, both French. He once again rode an Aprilia 250. Nothing changed from last year, with Guintoli again finishing in midpoints scoring positions. He finished the season in 10th place, with 84 points.
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In the case of three non-collinear points in the plane, the triangle with these points as its vertices has a unique Steiner inellipse that is tangent to the triangle's sides at their midpoints. The major axis of this ellipse falls on the orthogonal regression line for the three vertices.Minda and Phelps (2008), Corollary 2.4.
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Additionally, there are four more diagonal lines connecting the midpoints. These four additional diagonal lines is what makes the Picaria board different from Tapatan or Achi. The intersection points are where the pieces are played. The second version uses a similar board except there are four additional spaces or intersection points to play pieces at.
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It touches the sides at its midpoints. There is no other (non-degenerate) conic section with the same properties, because a conic section is determined by 5 points/tangents. b) By a simple calculation. c) The circumcircle is mapped by a scaling, with factor 1/2 and the centroid as center, onto the incircle.
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In astrology, a composite chart is a chart that is composed of the planetary midpoints of two or more horoscopes. Practitioners of astrology commonly construct a composite chart when two people meet and form a relationship. According to astrologers, the composite chart will give clues as to the nature and function of the relationship.
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This is an ideal titration curve for alanine, a diprotic amino acid. Point 2 is the first equivalent point where the amount of NaOH added equals the amount of alanine in the original solution. For each diprotic acid titration curve, from left to right, there are two midpoints, two equivalence points, and two buffer regions.
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Thermal unfolding curves and unfolding transition midpoints of two monoclonal antibodies. (A) Thermal unfolding curves of two monoclonal antibodies in the presence of 25mM Na-Citrate at different pH vlaues. Insets show the pH- dependence of the first unfolding midpoint (Tm1). (B) Dependence of Tm1 and Tm2 on the pH of the buffer of all tested conditions.
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One may find a variety of colors of connector nodes, but these all have the same purpose and design. At their midpoints, each of the yellow and red struts has an apparent twist. At these points, the cross-sectional shape reverses. This design feature forces the connector nodes on the ends of the strut to have the same orientation.
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A second common form of the Truchet tiles, due to , decorates each tile with two quarter-circles connecting the midpoints of adjacent sides. Each such tile has two possible orientations. We have such a tiling: 250px This type of tile has also been used in abstract strategy games Trax and the Black Path Game, prior to Smith's work.
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Especially true in the American two-party system, political parties want to maximize vote allocated to their candidate. Political parties will adjust their platform to comply with the median voters' demand. The Comparative Midpoints Model represents this idea best: Both political parties will get as close to the competing party's platform while preserving its own identity.
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The nine- point circle passes through these three midpoints; thus, it is the circumcircle of the medial triangle. These two circles meet in a single point, where they are tangent to each other. That point of tangency is the Feuerbach point of the triangle. Associated with the incircle of a triangle are three more circles, the excircles.
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As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse.Kalman, Dan.
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An indirect midpoint occurs when a stellar body makes an aspect to the midpoint of two other stellar bodies without a physical body at this midpoint.Aren Ober: Midpoint Interpretation Simplified, 2nd edition. Cotter Books, Cleveland Ohio, 2009. Midpoints were first used as Half-Sums by Ptolemy in the 2nd century, with the concepts of the 1st and 2nd harmonics.
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The better-known nine-point circle is an instance of Bôcher's conic. The nine-point hyperbola is another instance. Bôcher used the four points of the complete quadrangle as three vertices of a triangle with one independent point: :Given a triangle ABC and a point P in its plane, a conic can be drawn through the following nine points: :: the midpoints of the sides of ABC, :: the midpoints of the lines joining P to the vertices, and :: the points where these last named lines cut the sides of the triangle. The conic is an ellipse if P lies in the interior of ABC or in one of the regions of the plane separated from the interior by two sides of the triangle, otherwise the conic is a hyperbola.
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On some tetrahedral dice, three numbers are shown on each face. The number rolled is indicated by the number shown upright at all three visible faces—either near the midpoints of the sides around the base or near the angles around the apex. Another configuration places only one number on each face, and the rolled number is taken from the downward face.
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Downhill folding is predicted to occur under conditions of extreme native bias, i.e. at low temperatures or in the absence of denaturants. This corresponds to the type 0 scenario in the energy landscape theory. At temperatures or denaturant concentrations close to their apparent midpoints, proteins may switch from downhill to two-state folding, the type 0 to type 1 transition.
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Example of an inellipse In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and Brocard inellipse (see examples section). For any triangle there exist an infinite number of inellipses.
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To find the Spieker circle of a triangle, the medial triangle must first be constructed from the midpoints of each side of the original triangle. The circle is then constructed in such a way that each side of the medial triangle is tangent to the circle within the medial triangle, creating the incircle. This circle center is named the Spieker center.
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The buildings were placed around the edges of an immense square, and linked to each other by porticoes; no building stood in isolation. To speed connections between buildings, Ledoux introduced covered arcades that linked the midpoints of adjacent sides, forming a square within the square. Columns abounded. The buildings themselves were replete with them, and 144 Doric columns supported the covered arcades.
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Church Hill, also known as Timber Ridge Plantation, is a historic plantation house located near Lexington, Rockbridge County, Virginia. It was built circa 1848, and is a two-story, three bay, rectangular brick Greek Revival style dwelling. It has a one-story, rear kitchen ell. The house features stuccoed Doric order pilasters at the corners and midpoints of the long sides.
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Notice that this is an affine invariant construction since parallelism and midpoints are invariant under affine transformations. Consider the parabola given by the parametrisation . This has the equation The tangent line at γ(0) has the equation and so the parallel lines are given by for sufficiently small The line intersects the curve at The locus of midpoints is given by These form a line segment, and so the limiting tangent line to this line segment as we tend to γ(0) is just the line containing this line segment, i.e. the line In that case the affine normal line to the curve at γ(0) has the equation In fact, direct calculation shows that the affine normal vector at γ(0), namely ξ(0), is given by Davis, D. (2006), Generic Affine Differential Geometry of Curves in Rn, Proc.
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An increasing amount of the research of the Hamburg School revolved around work with astrological midpoints and use of the extra planets. Unfortunately, Witte and Rudolph were pursued by the Gestapo as enemies of the Third Reich. Alfred Witte committed suicide before being sent to a concentration camp, and Ludwig Rudolph was indeed interned, the Rulebook for Planetary Pictures banned and burned by the Nazis.
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For the 2003 season, Guintoli returned to the 250cc class, racing for the Campetella Racing, riding a private Aprilia, partnering Italian Franco Battaini. He showed some good pace, fighting consistently for midpoints scoring positions. He finished his first race of the year, at Suzuka in 9th place. He started to get more competitive with a 6th and a 5th place at Mugello and Le Mans respectively.
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The nine-point center of a triangle lies at the midpoint between the circumcenter and the orthocenter. These points are all on the Euler line. A midsegment (or midline) of a triangle is a line segment that joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to one half of that third side.
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The posts include randomly located tan- and brown- colored granite blocks. Similar blocks are at the top and midpoints of the half-walls. The half-walls are coped with recently installed rectangular cut and finished stone slabs that have rough cut edges protruding over the half- walls. An octagonal pyramid of three sequentially smaller cast concrete courses with slightly flared edges cap the posts.
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The program allows the creation of numerous objects which can be measured, and potentially used to solve hard math problems. The program allows the determination of the midpoints and midsegments of objects. Geometer's Sketchpad can measure lengths of segments, measures of angles, area, perimeter, etc. Some of the tools include a construct function, which allows the user to create objects in relation to selected objects.
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The triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It has (itself) the minimum symmetry all lattices have: points of inversion at each lattice point and at 7 more points for each lattice point: at the midpoints of the edges and the faces, and at the center points. It is the only lattice type that itself has no mirror planes.
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The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral (see below). The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.
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In geometry, the isotomic conjugate of a point P with respect to a triangle ABC is another point, defined in a specific way from P and ABC: If the base points of the lines PA, PB, and PC on the sides opposite A, B, and C are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic congugate of P.
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There are around 59 tooth rows in the upper jaw and 62 tooth rows in the lower jaw. The pectoral fins are moderately large and wide, with rounded tips. The dorsal fins have rounded apexes and are placed well back on the body, the first originating behind the midpoints of the small pelvic fins. The first dorsal fin is about twice as high as the second.
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Karl Wilhelm Feuerbach had previously observed that the three feet of the altitudes of a triangle and the three midpoints of its sides all lie on a single circle, but Terquem was the first to prove that this circle also contains the midpoints of the line segments connecting each vertex to the orthocenter of the triangle.. He also gave a new proof of Feuerbach's theorem that the nine-point circle is tangent to the incircle and excircles of a triangle. Terquem's other contributions to mathematics include naming the pedal curve of another curve, and counting the number of perpendicular lines from a point to an algebraic curve as a function of the degree of the curve. He was also the first to observe that the minimum or maximum value of a symmetric function is often obtained by setting all variables equal to each other.
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If the incircle is tangent to the bases at P and Q, then P, I and Q are collinear, where I is the incenter.J. Wilson, Problem Set 2.2, The University of Georgia, 2010, . The angles AID and BIC in a tangential trapezoid ABCD, with bases AB and DC, are right angles. The incenter lies on the median (also called the midsegment; that is, the segment connecting the midpoints of the legs).
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Let r and s be two ultraparallel lines. From any two distinct points A and C on s draw AB and CB' perpendicular to r with B and B' on r. If it happens that AB = CB', then the desired common perpendicular joins the midpoints of AC and BB' (by the symmetry of the Saccheri quadrilateral ACB'B). If not, we may suppose AB < CB' without loss of generality.
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Other variations have also been used. The SPAD S.XIII fighter, while appearing to be a two bay biplane, has only one bay, but has the midpoints of the rigging braced with additional struts, however these are not structurally contiguous from top to bottom wing. The Sopwith Strutter has a W shape cabane, however as it doesn't connect the wings to each other, it doesn't add to the number of bays.
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A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, the parallelogram formed by the midpoints of its sides, is a rhombus. An equivalent condition is that the bimedians of the quadrilateral (the diagonals of the Varignon parallelogram) are perpendicular. A convex quadrilateral with diagonal lengths p and q and bimedian lengths m and n is equidiagonal if and only if :pq=m^2+n^2.
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Using gyrotrigonometry, a gyrovector addition can be found which operates according to the gyroparallelogram law. This is the coaddition to the gyrogroup operation. Gyroparallelogram addition is commutative. The gyroparallelogram law is similar to the parallelogram law in that a gyroparallelogram is a hyperbolic quadrilateral the two gyrodiagonals of which intersect at their gyromidpoints, just as a parallelogram is a Euclidean quadrilateral the two diagonals of which intersect at their midpoints.
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In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle. A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base and summit and so either half of the Saccheri quadrilateral is a Lambert quadrilateral.
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The theorem was further generalized by Dao Thanh Oai. The generalization as follows: First generalization: Let ABC be a triangle, P be a point on the plane, let three parallel segments AA', BB', CC' such that its midpoints and P are collinear. Then PA', PB', PC' meet BC, CA, AB respectively at three collinear points. Dao's second generalization Second generalization: Let a conic S and a point P on the plane.
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In the figure, \scriptstyle R_x is the fixed, yet unknown, resistance to be measured. \scriptstyle R_1, \scriptstyle R_2, and \scriptstyle R_3 are resistors of known resistance and the resistance of \scriptstyle R_2 is adjustable. The resistance \scriptstyle R_2 is adjusted until the bridge is "balanced" and no current flows through the galvanometer \scriptstyle V_g. At this point, the voltage between the two midpoints (B and D) will be zero.
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The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex), then the area of the parallelogram is half the area of the quadrilateral. If one introduces the concept of oriented areas for n-gons, then this area equality also holds for complex quadrilaterals.Coxeter, H. S. M. and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited.
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Ledoux's plan envisaged that the central square courtyard would be where the factory would keep its firewood. At each corner of the square, and at the midpoints of each side stood two-story, square buildings that would house the various parts of the operation. In front were the quarters for the guards, a chapel, and a bakery. On the sides were workshops for the coopers and other workmen.
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In this case the vertices of A1 are the free apices of isosceles triangles with apex angles π/2 erected over the sides of the quadrilateral A0. The vertices of the quadrilateral A2 are the midpoints of the sides of the quadrilateral A1. By the PDN theorem, A2 is a square. The vertices of the quadrilateral A1 are the centers of squares erected over the sides of the quadrilateral A0.
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Not all natal points are created equal. If a planet happens to be at the center of a lot of midpoints, then transits to that point will be more powerful. A midpoint refers to the arithmetic mean of the zodiac degrees of two planets. Say, in a particular chart, Venus is at 0° Aries, Mercury is at 0° Taurus, and Mars is at 0° Gemini, then Mercury is at the midpoint of Venus/Mars.
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To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints of B'D and D'B respectively, and let F and H be the centers of the two squares. Then the theorem states that the quadrilateral EFGH is a square as well.. The square EFGH is called the Finsler–Hadwiger square of the two given squares.. See problem 8, pp. 20–21.
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Alfred Witte (2 March 1878 in Hamburg, Germany – 4 August 1941Alfred Witte Biographie (German: Alfred Witte - Begründer der modernen Astrologie) in Hamburg, Germany), was a German surveyor, astrologer, an amateur astronomer, and the founder of the Hamburg School of Astrology. Witte revived and further developed the use of astrological midpoints (a+b)/2 for precision in astrological analysis and prediction. Alfred Witte died 4 August 1941, Hamburg. The time of death is unclear.
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The Ersatz Monarch-class ships would have been protected at the waterline with an armored belt measuring thick amidships. This armor belt was to be located between the midpoints of the fore and aft barbettes, and would have thinned to further towards the ends of the ships. Their deck would have been thick. The main-gun turrets were designed to have of armor, while the casemates would have been shielded by armor plates thick.
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A specific case of Reusch's theorem where all four vertices of a tetrahedron are coplanar and lie on a single plane, thereby degenerating into a quadrilateral, Varignon's theorem, named after Pierre Varignon, states the following:Coxeter, op. cit., S. 242DUDEN: Rechnen und Mathematik. 1985, S. 652 :Let a quadrilateral in \R^2 be given. Then the two midlines connecting opposite edge midpoints intersect in the centroid of the quadrilateral and are divided in half by it.
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The three lines connecting the excenters of the given triangle to the corresponding edge midpoints all meet at the mittenpunkt; thus, it is the center of perspective of the excentral triangle and the median triangle, with the corresponding axis of perspective being the trilinear polar of the Gergonne point.. The mittenpunkt is also the centroid of the Mandart inellipse of the given triangle, the ellipse tangent to the triangle at its extouch points..
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Proof: can be done (like the properties above) for the unit parabola y = x^2. Application: This property can be used to determine the direction of the axis of a parabola, if two points and their tangents are given. An alternative way is to determine the midpoints of two parallel chords, see section on parallel chords. Remark: This property is an affine version of the theorem of two perspective triangles of a non-degenerate conic.
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Suppose that the distance between points λ1(t) and λ2(t) are constant for each t ∈ R and that the curve defined by the midpoints between λ1 and λ2 is such that its tangent vector at the point t is parallel to the segment from λ1(t) to λ2(t) for each t. If the curves λ1 and λ2 parametrizes the same smooth closed curve, then this curve is a Zindler curve.
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Similar to the logarithmic scale one can have a double logarithmic scale (example provided here) and super-logarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the generalised f-mean with f(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (e.g.
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Since P is across side AC, the 9-point conic is a nine-point hyperbola in this instance. When P is inside triangle ABC the 9-point conic is a nine-point circle. In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle. The nine-point conic was described by Maxime Bôcher in 1892.
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The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle. The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle.
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In this case it is normally quantified by comparing the assays response to a range of similar analytes and expressed as a percentage. In practice, calibration curves are produced using fixed concentration ranges for a selection of related compounds and the midpoints (IC50) of the calibration curves are calculated and compared. The figure then provides an estimate of the response of the assay to possible interfering compounds relative to the target analyte.
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Rectangular orange panels are placed on the space between floors, as well as on the piers between architectural bays, while circular green panels are placed in the middle of the piers between each floor, at the midpoints between four rectangular panels. An intricately molded cornice is located above the 16th floor. The northern and western facades consist of white buff brick with windows on the 9th through 17th floors, and have very little ornamentation.
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As a map , a similarity of ratio takes the form :f(x) = rAx + t, where is an orthogonal matrix and is a translation vector. Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it. The similarities of Euclidean space form a group under the operation of composition called the similarities group .
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The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area. The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle. A parallelogram with equal diagonals is a rectangle. The Japanese theorem for cyclic quadrilateralsCyclic Quadrilateral Incentre-Rectangle with interactive animation illustrating a rectangle that becomes a 'crossed rectangle', making a good case for regarding a 'crossed rectangle' as a type of rectangle.
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As illustrated in the case where n = 8, the method consists of changing the position of the numbers that lie of the sides of the square that is formed by joining the midpoints of the sides of the mystic square (Figure 3). Each of these lines is first “reflected” with the number on the opposite end of the same line (Figure 4). These numbers are in turn reflected “across the board” (Figure 5). This produces an 8x8 Magic Square.
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The jaws are long and narrowly triangular, and can be protruded from the head. The teeth are distinctively fang-like and widely spaced; the most anterior teeth are grooved lengthwise. Six to 10 upper and seven to 10 lower tooth rows occur on each side, along with a single tooth row at the upper and lower symphyses (jaw midpoints). The teeth are largest at the symphysis and decline in size towards the corners of the mouth.
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Baba Jan III. On the Central Mound only the superimposed foundations from two phases of a Manor (30 x 35 m) survive. The earlier, level 2, consisted of a rectangular court with a north-south axis flanked by long narrow rooms, the whole with towers both at the corners and midpoints of all sides. When rebuilt in level 1, the Manor consisted of a central columned hall (one row of three columns) surrounded by long narrow rooms.
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All kites tile the plane by repeated inversion around the midpoints of their edges, as do more generally all quadrilaterals. A kite with angles π/3, π/2, 2π/3, π/2 can also tile the plane by repeated reflection across its edges; the resulting tessellation, the deltoidal trihexagonal tiling, superposes a tessellation of the plane by regular hexagons and isosceles triangles.See . The deltoidal icositetrahedron, deltoidal hexecontahedron, and trapezohedron are polyhedra with congruent kite-shaped facets.
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A titration curve for a diprotic acid contains two midpoints where pH=pKa. Since there are two different Ka values, the first midpoint occurs at pH=pKa1 and the second one occurs at pH=pKa2. Each segment of the curve which contains a midpoint at its center is called the buffer region. Because the buffer regions consist of the acid and its conjugate base, it can resist pH changes when base is added until the next equivalent points.
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The Eight Directions represent the four seasons (North – Winter, South – Summer, East – Spring, and West – Autumn) and the Winter and Summer Solstices, as well as the Spring and Fall Equinoxes. The midpoints between those four times of year are the four lesser directions. This Eight Direction model maps perfectly onto the eight arrows of the root chakra. The four petals of the chakra also map onto the four elements of Earth (North), Air (East), Fire (South) and Water (West).
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Also, the midpoints of each side of the medial triangle are found and connected to the midpoint of the opposite line through the Nagel point. Each of these lines share a common midpoint, S. With each of these lines reflected through S, the result is 6 points within the medial triangle. Draw a conic through any 5 of these reflected points and the conic will touch the final point. This was proven by de Villiers in 2006.
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Nine-point hyperbola: One branch bisects BA, BC, and BP. The other branch bisects PA, PC, and AC, as well as passing through BA.PC and AP.BC. In plane geometry with triangle ABC, the nine-point hyperbola is an instance of the nine-point conic described by Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic: :Given a triangle ABC and a point P in its plane, a conic can be drawn through the following nine points: :: the midpoints of the sides of ABC, :: the midpoints of the lines joining P to the vertices, and :: the points where these last named lines cut the sides of the triangle. The conic is an ellipse if P lies in the interior of ABC or in one of the regions of the plane separated from the interior by two sides of the triangle; otherwise, the conic is a hyperbola. Bôcher notes that when P is the orthocenter, one obtains the nine-point circle, and when P is on the circumcircle of ABC, then the conic is an equilateral hyperbola.
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For a convex quadrilateral with sides a, b, c, d, diagonals e and f, and g being the line segment connecting the midpoints of the two diagonals, the following equations holds: :a^2+b^2+c^2+d^2=e^2+f^2+4g^2 If the quadrilateral is a parallelogram, then the midpoints of the diagonals coincide so that the connecting line segment g has length 0. In addition the parallel sides are of equal length, hence Euler's theorem reduces to :2a^2+2b^2=e^2+f^2 which is the parallelogram law. If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: :2a^2+2b^2=2e^2 Dividing by 2 yields the Euler–Pythagoras theorem: :a^2+b^2=e^2 In other words, in the case of a rectangle the relation of the quadrilateral's sides and its diagonals is described by the Pythagorean theorem.Lokenath Debnath: The Legacy of Leonhard Euler: A Tricentennial Tribute.
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The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, and hence are coplanar.
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The upper and lower jaws contain 14–15 and 12–14 tooth rows on either side respectively; in addition, there are one or two rows of small teeth at the upper and lower symphyses (jaw midpoints). The upper teeth have a single narrow, smooth-edged central cusp, flanked on both sides by very large serrations. The lower teeth are narrower and more upright than the uppers, and may be smooth to finely serrated. The five pairs of gill slits are fairly long.
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The mittenpunkt M of the black triangle, at the center of its Mandart inellipse (red). The blue lines through the middenpunkt pass through the triangle's excenters and corresponding edge midpoints. In geometry, the mittenpunkt (German, middlespoint) of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It was identified in 1836 by Christian Heinrich von Nagel as the symmedian point of the excentral triangle of the given triangle...
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However, other elements of the design including the roof and second-story window treatments have a character more reminiscent of the Renaissance Revival architectural style of the late 1880s. To the north of the original building is a 1929-30 addition. Both buildings are rectangular, and are oriented parallel to each other, with the addition being slightly smaller than the original building. A two-story vestibule, built at the same time as the addition, connects the buildings at their midpoints.
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The incircle of a triangle ABC is a circle that is tangent to all three sides of the triangle. Its center, the incenter of the triangle, lies at the point where the three internal angle bisectors of the triangle cross each other. The nine-point circle is another circle defined from a triangle. It is so called because it passes through nine significant points of the triangle, among which the simplest to construct are the midpoints of the triangle's sides.
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Astroid The hypocycloid construction of the astroid. Astroid x^{2/3}+y^{2/3}=r^{2/3} as the common envelope of a family ellipses of equation (x/a)^2+(y/b)^2=r^2, where a + b = 1. The envelope of a ladder (coloured lines in the top-right quadrant) sliding down a vertical wall, and its reflections (other quadrants) is an astroid. The midpoints trace out a circle while other points trace out ellipses similar to the previous figure.
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The circular eyes, located at the forward outer corners of the cephalofoil, are equipped with protective nictitating membranes. The relatively small, arched mouth contains 15–16 upper and 14 lower tooth rows on each side, and sometimes also a single row of tiny teeth at the upper and/or lower symphyses (jaw midpoints). The teeth are small and smooth-edged, with angled triangular cusps. Five pairs of gill slits are seen, with the fifth pair over the pectoral fin origins.
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However, under pressure from the Minister, the commission selected the iron dome on 20 August 1807. Construction of the iron dome covered in sheets of copper began in 1809 and was completed in 1811. The engineer François Brunet assisted Bélanger in the calculations and design of the dome, which had a diameter of more than . It was made of 51 sections, corresponding to the midpoints of the 25 arches of the rotunda, with each section made of two beams connected by spacers.
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A midpoint is a mathematical point halfway between two stellar bodies that tells an interpretative picture for the individual. There are two types of midpoints: direct and indirect. A direct midpoint occurs when a stellar body makes an aspect to the midpoint of two other stellar bodies with an actual physical body at the halfway point. In other words, a direct midpoint means that there is actually a stellar body in the natal chart lying in the midpoint of two other stellar bodies.
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After the AIEE's founding in 1884, its member's badge was created in 1893 by a committee headed by Dr. Alexander Graham Bell, President of AIEE from 1891 to 1892. The badge's logo depicted Benjamin Franklin's kite, representative of the discovery that lightning carried electricity. The design also showed a winding of gold wire with its midpoints crossed by a galvanometer's indicator, invoking the electrical engineer's Wheatstone bridge. Additionally, Ohm's law and the letters 'AIEE' were added in gold at the logo's base.
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The pigeye shark is a very robust-bodied species with a short, broad, and rounded snout. The small and circular eyes are equipped with nictitating membranes. The anterior rims of the nostrils bear medium-sized flaps of skin. The mouth forms a wide arch and has barely noticeable furrows at the corners. There are 11–13 (usually 12) upper and 10–12 (usually 11) lower tooth rows on each side; in addition, there are single rows of tiny teeth at the upper and lower symphyses (jaw midpoints).
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After the buttresses were completed, the construction crews could then begin building the falsework on which the concrete would harden. While the concrete was drying, the roof was supported by a central steel tower, the buttresses around the periphery, a circular timber tower about from the center, and two movable timber scaffolds at the midpoints of and -span joists. These supports totaled at of wooden scaffolding. To aid in construction of the roof, a temporary steel tower was erected in the center of the stadium.
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The pseudomedian of a distribution F is defined to be a median of the distribution of (Z_1+Z_2)/2, where Z_1 and Z_2 are independent, each with the same distribution F.Hollander, M. and Wolfe, D. A. (2014). Nonparametric Statistical Methods (3nd Ed.). p58 When F is a symmetric distribution, the pseudomedian coincides with the median, otherwise this is not generally the case. The Hodges–Lehmann statistic, defined as the median of all of the midpoints of pairs of observations, is a consistent estimator of the pseudomedian.
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This structural topology is described as 51234. A short (two to four turns) N-terminal alpha helix is also present in most LSm proteins. The β3 and β4 strands are short in some LSm proteins, and are separated by an unstructured coil of variable length. The β2, β3 and β4 strands are strongly bent about 120° degrees at their midpoints The bends in these strands are often glycine, and the side chains internal to the beta barrel are often the hydrophobic residues valine, leucine, isoleucine and methionine.
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Similar hinges were located at the midpoints of the long horizontal bars of the rectangle. These midpoint pivots connected to the vertical bar on the table. The result was a pantograph that allowed the long horizontal bars to be rotated into the vertical to point upward at an aircraft, sighting along the upper bar. A final piece was a separate vertical bar connected to the two horizontals and pivoted in the same way so that it remained pointing vertically as the horizontal bars were rotated.
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Additionally, there are twelve octahedral voids located at the midpoints of the edges of the unit cell as well as one octahedral hole in the very center of the cell, for a total of four net octahedral voids. One important characteristic of a crystalline structure is its atomic packing factor. This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres.
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This consists in avoiding to round to midpoints for the final rounding (except when the midpoint is exact). In binary arithmetic, the idea is to round the result toward zero, and set the least significant bit to 1 if the rounded result is inexact; this rounding is called sticky rounding. Equivalently, it consists in returning the intermediate result when it is exactly representable, and the nearest floating-point number with an odd significand otherwise; this is why it is also known as rounding to odd.
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This arrangement allowed J.D. Irving Ltd. to use New Brunswick Railway Co. Ltd. as a holding company to own both the NBSR as well as its U.S. sister EMRY. Ownership of the tracks in New Brunswick extends to the Canada–United States border at the midpoints of two crossings of the St. Croix River; these being the Saint Croix-Vanceboro Railway Bridge at St. Croix shared with the EMRY, as well as an unnamed railway bridge at St. Stephen shared with Pan Am Railways .
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In 2015, West was partnered this time by a more competitive rider than the others, Spain's Julián Simón. This time West's performance were not so good. He often struggled in qualifying and he always had to regain positions in the race, going to the limits and retiring more often than previous years. Tensions with the team didn't help and when West was finally in the midpoints positions, he was sacked by the team after the San Marino GP. He was replaced by Mika Kallio.
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The triples of red points on the two black lines have the same distances within each triple, so by Hjelmslev's theorem the three midpoints of corresponding pairs of points are on a single (green) line. In geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... on a line are isometrically mapped to points P´, Q´, R´... of another line in the same plane, then the midpoints of the segments PP´, QQ´, RR´... also lie on a line. The proof is easy if one assumes the classification of plane isometries. If the given isometry is odd, in which case it is necessarily either a reflection in a line or a glide-reflection (the product of three reflections in a line and two perpendiculars to it), then the statement is true of any points in the plane whatsoever: the midpoint of PP´ lies upon the axis of the (glide-)reflection for any P. If the isometry is even, compose it with reflection in line PQR to obtain an odd isometry with the same effect on P, Q, R... and apply the previous remark.
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Then as a variable line lies on the point, find the locus of the midpoint of the segment determined by the planes. Young's solution starts with a line p through the point and parallel to the intersection of the planes. She identified the locus as a hyperbolic cylinder through use of a third parallel midway between the others that is the projective harmonic conjugate of a line at infinity.AMM 31(7): 356 In a triangle ABC the feet of the altitudes and midpoints of the sides are used to define three involutions.
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The medial triangle of a given triangle has vertices at the midpoints of the given triangle's sides, therefore its sides are the three midsegments of the given triangle. It shares the same centroid and medians with the given triangle. The perimeter of the medial triangle equals the semiperimeter (half the perimeter) of the original triangle, and its area is one quarter of the area of the original triangle. The orthocenter (intersection of the altitudes) of the medial triangle coincides with the circumcenter (center of the circle through the vertices) of the original triangle.
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Every Paley graph is self-complementary. For example, the 3 × 3 rook's graph (the Paley graph of order nine) is self-complementary, by a symmetry that keeps the center vertex in place but exchanges the roles of the four side midpoints and four corners of the grid.. All strongly regular self-complementary graphs with fewer than 37 vertices are Paley graphs; however, there are strongly regular graphs on 37, 41, and 49 vertices that are not Paley graphs.. The Rado graph is an infinite self-complementary graph.. See in particular Proposition 5.
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Many of the buildings visible from the trail date back to this period of Victorian architecture. Additional focal points/parking lots for the trail are located at midpoints at the Michigan Iron Industry Museum in Negaunee and the Cliffs Shaft Mine Museum in Ishpeming. The Iron Ore Heritage Trail is operated by the Iron Ore Heritage Trail Recreation Authority, a free-standing unit of local government that collects a property tax to support the trail. The trail was largely completed in 2013, but has been upgraded in several cycles since that time.
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A polygon is defined to be cyclic if its vertices are all concyclic. For example, all the vertices of a regular polygon of any number of sides are concyclic. A tangential polygon is one having an inscribed circle tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle. A convex quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four altitudes are eight concyclic points, on what is called the eight-point circle.
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They can be found by fitting a straight line using least squares technique for a small window of neighboring pixels. The next step is to find the intersection points of the tangent lines. This can be easily done by solving the line equations found in the previous step. Then let the intersection points be T12 and T23, the midpoints of line segments X_1X_2 and X_2X_3 be M12 and M23. Then the center of the ellipse will lie in the intersection of T_{12}M_{12} and T_{23}M_{23}.
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The annual cycle of insolation (Sun energy, shown in blue) with key points for seasons (middle), quarter days (top) and cross-quarter days (bottom) along with months (lower) and Zodiac houses (upper). The cycle of temperature (shown in pink) is delayed by seasonal lag. Solar timing is based on insolation in which the solstices and equinoxes are seen as the midpoints of the seasons. It was the method for reckoning seasons in medieval Europe, especially by the Celts, and is still ceremonially observed in Ireland and some East Asian countries.
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A cubic polynomial has three zeroes in the complex number plane, which in general form a triangle, and the Gauss–Lucas theorem states that the roots of its derivative lie within this triangle. Marden's theorem states their location within this triangle more precisely: :Suppose the zeroes , , and of a third-degree polynomial are non-collinear. There is a unique ellipse inscribed in the triangle with vertices , , and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative .
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There are an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.
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The H-101 differs from the Libelle in having a V-tail, showing its ancestry to the V-tailed Hütter H-30 GFK. Four flush-fitting air brakes were fitted to the trailing edges of the wings, replacing the more conventionally sited air brakes of the Standard Libelle. The Salto's air brakes are hinged at their midpoints so that half the surface projects above the wing and half below. The Salto prototype first flew on 6 March 1970, and 67 had been delivered by early 1977, when production at Start + Flug GmbH Saulgau ceased.
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Szent István was protected at the waterline with an armor belt which measured thick in the central citadel, where the most- important parts of the ship were located. This armor belt was located between the midpoints of the fore and aft barbettes, and thinned to further towards the bow and stern, but did not reach either. It was continued to the bow by a small patch of armor. The upper armor belt had a maximum thickness of , but it thinned to from the forward barbette all the way to the bow.
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Texels can also be described by image regions that are obtained through simple procedures such as thresholding. Voronoi tesselation can be used to define their spatial relationships—divisions are made at the midpoints between the centroids of each texel and the centroids of every surrounding texel for the entire texture. This results in each texel centroid having a Voronoi polygon surrounding it, which consists of all points that are closer to its own texel centroid than any other centroid.Linda G. Shapiro and George C. Stockman, Computer Vision, Upper Saddle River: Prentice-Hall, 2001.
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Since Minos controlled the land and sea routes, Daedalus set to work to fabricate wings for himself and his young son Icarus. He tied feathers together, from smallest to largest so as to form an increasing surface. He secured the feathers at their midpoints with string and at their bases with wax, and gave the whole a gentle curvature like the wings of a bird. When the work was done, the artist, waving his wings, found himself buoyed upward and hung suspended, poising himself on the beaten air.
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The Tegetthoff-class ships were protected at the waterline with an armor belt which measured thick in the central citadel, where the most important parts of the ship were located. This armor belt was located between the midpoints of the fore and aft barbettes, and thinned to further towards the bow and stern, but did not reach either. It was continued to the bow by a small patch of armor. The upper armor belt had a maximum thickness of , but it thinned to from the forward barbette all the way to the bow.
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The resulting effect is the same as convolving with a two- dimensional kernel in a single pass, but requires fewer calculations. Discretization is typically achieved by sampling the Gaussian filter kernel at discrete points, normally at positions corresponding to the midpoints of each pixel. This reduces the computational cost but, for very small filter kernels, point sampling the Gaussian function with very few samples leads to a large error. In these cases, accuracy is maintained (at a slight computational cost) by integration of the Gaussian function over each pixel's area.
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400px We assume that P is not collinear with any two vertices of ABC. Let A', B' and C' be the points in which the lines AP, BP, CP meet sidelines BC, CA and AB (extended if necessary). Reflecting A', B', C' in the midpoints of sides BC, CA, AB will give points A", B" and C" respectively. The isotomic lines AA", BB" and CC" joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of P.
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The Wheel of the Year in the Northern Hemisphere. Pagans in the Southern Hemisphere advance these dates six months to coincide with their own seasons. The Wheel of the Year is an annual cycle of seasonal festivals, observed by many modern Pagans, consisting of the year's chief solar events (solstices and equinoxes) and the midpoints between them. While names for each festival vary among diverse pagan traditions, syncretic treatments often refer to the four solar events as "quarter days" and the four midpoint events as "cross-quarter days", particularly in Wicca.
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First row (square): 00 10 01 11 Second row : 000 100 010 001 triad (triangle) 110 101 011 triad (triangle) 111 Third row 0000 1000 0100 0010 0001 tetrad (tetrahedron or 3-simplex) 1100 1010 1001 0110 0101 0011 hexany (octahedron) 1110 1101 1011 0111 tetrad 1111 The octahedron there is the edge dual of the tetrahedron, or rectified tetrahedron Fourth row 00000 10000 01000 00100 00010 00001 pentad (4-simplex or pentachoron – four- dimensional tetrahedron) 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 2)5 dekany (10 vertices, rectified 4-simplex) 00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 3)5 dekany (10 vertices) 01111 10111 11011 11101 11110 pentad 11111 The rectified 4-simplex which is the mathematical name for the geometrical shape of the dekany is also known as the dispentachoron Fifth row 000000 100000 010000 001000 000100 000010 000001 hexad (5-simplex or hexateron – five-dimensional tetrahedron) 110000 101000 100100 100010 100001 011000 010100 010010 010001 001100 001010 001001 000110 000101 000011 2)6 pentadekany (15 vertices, rectified 5-simplex) 111000 110100 110010 110001 101100 101010 101001 100110 100101 100011 011100 011010 011001 010110 010101 010011 001110 001101 001011 000111 eikosany (20 vertices birectified 5-simplex) 001111 010111 011011 011101 011110 100111 101011 101101 101110 110011 110101 110110 111001 111010 111100 4)6 pentadekany (15 vertices) 011111 101111 110111 111011 111101 111110 hexad 111111 The dekany is the edge dual of the 4-simplex. Similarly, the geometrical figure for the pentadekany is the edge dual of the 5-simplex. A dekany cam be made by joining together the midpoints of the edges of the 4-simplex, and similarly for the pentadekany and the 5-simplex. Similarly the dekany vertices when scaled by 1/2 move to the midpoints of the 4-simplex edges, and the pentadekany vertices move to the midpoints of the 5-simplex edges, and so on in all higher dimensions.
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The "Orion 606" SM design retained the width for the attachments of the Orion M with the Orion CM, but utilized a Soyuz-like service module design to allow Lockheed Martin to make the vehicle lighter in weight and permitting the attachment of the circular solar panels at the module's midpoints, instead of at the base near the spacecraft/rocket adapter, which might have subjected the panels to damage. The Orion service module (SM) was projected comprising a cylindrical shape, having a diameter of and an overall length (including thruster) of . The projected empty mass was , fuel capacity was .
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Every triangle has an inscribed ellipse, called its Steiner inellipse, that is internally tangent to the triangle at the midpoints of all its sides. This ellipse is centered at the triangle's centroid, and it has the largest area of any ellipse inscribed in the triangle. In a right triangle, the circumcenter is the midpoint of the hypotenuse. In an isosceles triangle, the median, altitude, and perpendicular bisector from the base side and the angle bisector of the apex coincide with the Euler line and the axis of symmetry, and these coinciding lines go through the midpoint of the base side.
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Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the cubic equation \scriptstyle (x-a)(x-b)(x-c)=0, take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.
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Salary range minimums were being aligned to approximately the 25th percentile of the market data, midpoints were being aligned to market medians, and maximums were being aligned to the 75th percentile of the market data. In an article published by WorldatWork, Stoskopf first coined the term "market-based salary structure" to describe this phenomenon. In 2012, WorldatWork and Deloitte Consulting LLP co-sponsored research to determine the extent to which this practice had become prevalent in US employers' salary administration practices. By that time 64% of employers utilized market-based salary structures to administer pay for their employees.
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The sums of the areas of opposing triangles are equal, that is Area(BCL) + Area(DAL) = Area(LAB) + Area(DLC) Anne's theorem, named after the French mathematician Pierre-Leon Anne (1806–1850), is a statement from Euclidean geometry, which describes an equality of certain areas within a convex quadrilateral. Specifically, it states: :Let ABCD be a convex quadrilateral with diagonals AC and BD, that is not a parallelogram. Furthermore let E and F be the midpoints of the diagonals and L be an arbitrary point in the interior of ABCD. L forms four triangles with the edges of ABCD.
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A rectified cube is a cuboctahedron – edges reduced to vertices, and vertices expanded into new faces A birectified cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices. A rectified cubic honeycomb – edges reduced to vertices, and vertices expanded into new cells. In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.
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The octahedron represents the central intersection of two tetrahedra The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.
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This arrangement has only m ordinary lines, the lines that connect a vertex v with the point at infinity collinear with the two neighbors of v. As with any finite configuration in the real projective plane, this construction can be perturbed so that all points are finite, without changing the number of ordinary lines. For odd n, only two examples are known that match Dirac's lower bound conjecture, that is, with t_2(n)=(n-1)/2 One example, by , consists of the vertices, edge midpoints, and centroid of an equilateral triangle; these seven points determine only three ordinary lines.
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The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles. Euler's line is a straight line through the orthocenter (blue), center of the nine- point circle (red), centroid (orange), and circumcenter (green) The orthocenter (blue point), center of the nine-point circle (red), centroid (orange), and circumcenter (green) all lie on a single line, known as Euler's line (red line).
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A winghead shark long is theoretically capable of sampling over of water per second. Another potential olfactory benefit of the cephalofoil is increased separation between the midpoints of the left and right nostrils, which enhances the shark's ability to resolve the direction of a scent trail. Finally, the cephalofoil may increase the shark's ability to detect the electric fields and movements of its prey, by providing a larger surface area for its electroreceptive ampullae of Lorenzini and mechanoreceptive lateral line. The lateral blades seem too large to function in maneuvering, which has been suggested for other hammerheads.
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Wallace Sewell moquette upholstery Class 378 in revised London Overground livery All Electrostar and Aventra stock in service now carries Overground livery. It is similar to Underground livery, and consists of white and black coaches, a longitudinal thick blue stripe and a thin orange stripe along the bottom, London Overground roundels at midpoints along the coaches, black window-surrounds and orange doors. The ends of each unit are painted yellow to comply with the National Rail standards that existed when the first wave of new trains began to enter service in 2009. The seat upholstery features a moquette by fabric designers Wallace Sewell.
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It was intended as the first of a range of similar Emsco aircraft, differing in having one or two engines. The Challenger's wing was built in two parts, both rectangular in plan out to semi-elliptical tips, which met on top of the fuselage and were mounted with 1.5° dihedral. They had wooden structures built around two box spars and were fabric covered. Parallel struts from beyond mid- span braced the spars to the lower fuselage longerons and the rear struts were also braced near their midpoints to the upper longerons; all struts were enclosed in wide, airfoil section fairings.
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A regular polygon has an inscribed circle which is tangent to each side of the polygon at its midpoint. In a regular polygon with an even number of sides, the midpoint of a diagonal between opposite vertices is the polygon's center. The midpoint-stretching polygon of a cyclic polygon (a polygon whose vertices all fall on the same circle) is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of .. Iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shapes converge to that of a regular polygon.
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The vertices of the first stellation of the rhombic dodecahedron include the 12 vertices of the cuboctahedron, together with eight additional vertices (the degree-3 vertices of the rhombic dodecahedron). Escher's solid has six additional vertices, at the center points of the square faces of the cuboctahedron (the degree-4 vertices of the rhombic dodecahedron). In the first stellation of the rhombic dodecahedron, these six points are not vertices, but are instead the midpoints of pairs of edges that cross at right angles at these points. The first stellation of the rhombic dodecahedron has 12 hexagonal faces, 36 edges, and 20 vertices, yielding an Euler characteristic of 20 − 36 + 12 = −4.
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The term most frequently refers to the school of astrology founded by Ebertin. The main difference between Witte's Hamburg School and Ebertin's Cosmobiology is that Cosmobiology rejects the hypothetical Trans-Neptunian objects used by the Hamburg School and practitioners of Uranian astrology. Another difference is the significant expansion of Cosmobiology into medical astrology, Dr. Ebertin being a physician. Cosmobiology continued Witte's ultimate primary emphasis on the use of astrological midpoints along with the following 8th-harmonic aspects in the natal chart, which both Witte and Ebertin found to be the most potent in terms of personal influence: conjunction (0°), semi-square (45°), square (90°), sesquiquadrate (135°), and opposition (180°).
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The eight-cylinder 21–60 held the vertical shaft in the centre of the engine, and both crankshaft and camshaft were divided at their midpoints. Their smallest engine of 847cc was designed and made for Morris's new Minor at Ward End with the camshaft drive's shaft the spindle of the dynamo driven by spiral bevel gears. But it was relatively expensive to build and inclined to oil leaks, so its design was modified to a conventional side-valve layout by Morris Engines, which was put into production just for Morris cars in 1932. Meanwhile, Wolseley expanded their original design from four to six cylinders.
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If the two sums of areas of opposite triangles are equal ( Area(BCL) + Area(DAL) = Area(LAB) + Area(DLC) ), then the point L is located on the Newton line, that is the line which connects E and F. For a parallelogram the Newton line does not exist since both midpoints of the diagonals coincide with point of intersection of the diagonals. Moreover the area identity of the theorem holds in this case for any inner point of the quadrilateral. The converse of Anne's theorem is true as well, that is for any point on the Newton line which is an inner point of the quadrilateral, the area identity holds.
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The dessin d'enfant on the Klein quartic associated with the quotient map by its automorphism group (with quotient the Riemann sphere) is precisely the 1-skeleton of the order-3 heptagonal tiling.. That is, the quotient map is ramified over the points , and ; dividing by 1728 yields a Belyi function (ramified at , and ), where the 56 vertices (black points in dessin) lie over 0, the midpoints of the 84 edges (white points in dessin) lie over 1, and the centers of the 24 heptagons lie over infinity. The resulting dessin is a "platonic" dessin, meaning edge- transitive and "clean" (each white point has valence 2).
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If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by Steinitz's theorem it can be represented as a polyhedron. The Conway polyhedron notation equivalent to rectification is ambo, represented by a.
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The current laser at the UK Atomic Weapons Establishment (AWE), the HELEN (High Energy Laser Embodying Neodymium) 1-terawatt neodymium-glass laser, can access the midpoints of pressure and temperature regions and is used to acquire data for modeling on how density, temperature, and pressure interact inside warheads. HELEN can create plasmas of around 106 K, from which opacity and transmission of radiation are measured. Neodymium glass solid-state lasers are used in extremely high power (terawatt scale), high energy (megajoules) multiple beam systems for inertial confinement fusion. Nd:glass lasers are usually frequency tripled to the third harmonic at 351 nm in laser fusion devices.
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The line runs between the mainland portion of Gyeonggi-do province that had been part of Hwanghae before 1945, and the adjacent offshore islands, including Yeonpyeong and Baengnyeongdo. Because of the conditions of the armistice, the mainland portion reverted to North Korean control, while the islands remained a part of South Korea despite their close proximity. The line extends into the sea from the Military Demarcation Line (MDL), and consists of straight line segments between 12 approximate channel midpoints, extended in an arc to prevent egress between both sides. On its western end the line extends out along the 38th parallel to the median line between Korea and China.
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From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle. The Gergonne triangle or intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.
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The city was carefully planned during the early Ming dynasty, with a central axis running through the midpoints of the northern and southern walls, dividing the city into two equal parts to fulfil the Confucian ideals of "居中不偏" and "不正不威". The Forbidden city was at the centre of it all, with Longevity Mount (Jingshan) just north of the palaces. The Imperial city, surrounding the palaces, provided everything necessary for the maintenance of the Emperor's lifestyle within the Forbidden city. The southern sections of the city were allocated to officials and bureaucrats, while the Inner and Outer city served as extra protection for the Imperial city and the palace complex.
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A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in this way from the three pairs of opposite cube squares form the compound of three octahedra.. The eight cube vertices are the same as the eight points in the compound where three edges cross each other. Each of the octahedron edges that participates in these triple crossings is divided by the crossing point in the ratio 1:. The remaining octahedron edges cross each other in pairs, within the interior of the compound; their crossings are at their midpoints and form right angles.
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The route length of a transport network is the sum of the lengths of all routes in the network, such as railways, road sections or air sectors. The U.S. Department of Transportation's Federal Transit Administration has also referred to this as "Directional Route Miles (DRM)". Where a network is made up of railways, route length has also been defined, by at least one source, as the sum of the distances (in kilometres) between the midpoints of all stations on the network. In a measurement of route length, each route is counted only once, regardless of how many lines pass over it, and regardless of whether it is single track or multi track, single carriageway or dual carriageway.
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Click here for an animated version. The Herschel graph is planar and 3-vertex- connected, so it follows by Steinitz's theorem that it is a polyhedral graph: there exists a convex polyhedron (an enneahedron) having the Herschel graph as its skeleton.. This polyhedron has nine quadrilaterals for faces, which can be chosen to form three rhombi and six kites. Its dual polyhedron is a rectified triangular prism, formed as the convex hull of the midpoints of the edges of a triangular prism. This polyhedron has the property that its faces cannot be numbered in such a way that consecutive numbers appear on adjacent faces, and such that the first and last number are also on adjacent faces.
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He then built off Napoleon by proving that if an equilateral triangle was constructed with equilateral triangles incident on each vertex, the midpoints of the connecting lines between the non-incident vertices of the outer three equilateral triangles create an equilateral triangle. Other similar work was done by the French Geometer Thébault in his proof that given a parallelogram and squares that lie on each side of the parallelogram, the centers of the squares create a square. Mauldon then analyzed coplanar sets of triangles, determining if they were similarity systems based on the criterion, if all but one of the triangles were directly similar, then all of the triangles are directly similar.
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The kites are the quadrilaterals that have an axis of symmetry along one of their diagonals.. Any non-self-crossing quadrilateral that has an axis of symmetry must be either a kite (if the axis of symmetry is a diagonal) or an isosceles trapezoid (if the axis of symmetry passes through the midpoints of two sides); these include as special cases the rhombus and the rectangle respectively, which have two axes of symmetry each, and the square which is both a kite and an isosceles trapezoid and has four axes of symmetry. If crossings are allowed, the list of quadrilaterals with axes of symmetry must be expanded to also include the antiparallelograms.
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The van Lamoen circle through six circumcenters A_b, A_c, B_c, B_a, C_a, C_b In Euclidean plane geometry, the van Lamoen circle is a special circle associated with any given triangle T. It contains the circumcenters of the six triangles that are defined inside T by its three medians. Specifically, let A, B, C be the vertices of T, and let G be its centroid (the intersection of its three medians). Let M_a, M_b, and M_c be the midpoints of the sidelines BC, CA, and AB, respectively. It turns out that the circumcenters of the six triangles AGM_c, BGM_c, BGM_a, CGM_a, CGM_b, and AGM_b lie on a common circle, which is the van Lamoen circle of T.
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The four "maltitudes" of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is cyclic (inscribed in a circle), these maltitudes all meet at a common point called the "anticenter". Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always goes through the midpoint of the opposite side. Varignon's theorem states that the midpoints of the sides of an arbitrary quadrilateral form the vertices of a parallelogram, and if the quadrilateral is not self-intersecting then the area of the parallelogram is half the area of the quadrilateral.
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The MLFMM is based on the Method of Moments (MoM), but reduces the memory complexity from N2 to NlogN, and the solving complexity from N3 to NiterNlogN, where N represents the number of unknowns and Niter the number of iterations in the solver. This method subdivides the Boundary Element mesh into different clusters and if two clusters are in each other's far field, all calculations that would have to be made for every pair of nodes can be reduced to the midpoints of the clusters with almost no loss of accuracy. For clusters not in the far field, the traditional BEM has to be applied. That is MLFMM introduces different levels of clustering (clusters made out of smaller clusters) to additionally enhance computation speed.
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Michael Ahr of Den of Geek felt the failures endured in the episode made "for the perfect season midpoints as the characters face the most overwhelming odds yet, framing what will presumably be their seemingly impossible comeback in the back half", giving the episode 4.5 stars out of 5. Trent Moore at Syfy Wire felt Coulson's line "Dying? It’s kind of my superpower." was "one of the show’s greatest" and that Sousa and Daisy had "a great bit of chemistry" in their scenes, noting "Sousa really gets a chance to shine". Matt Webb Mitovich from TVLine said that after the first five episodes had "light-hearted time-travel hijinks", "Adapt or Die" had "harder-hitting drama" which resulted in an "incredibly eventful episode".
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The delimitation border Negotiations on the outside marine border were initiated in 1970. Norway claimed, in accordance with the United Nations Convention on the Law of the Sea Article 15 and the Convention on the High Seas, that the border should follow the equidistance principle, the border being defined by midpoints between the nearest land area or islands, as is normal practice internationally. The Soviet Union claimed, based on a decision by Joseph Stalin from 1926, which was not recognized by any other country than the Soviet Union, that a "sector principle" should apply, such that the border should follow meridian lines. Most of the disputed area was within what would normally be considered Norwegian according to the relevant international treaties.
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Using the notation in the diagram on the right, the sides are (AB), (BC), (CD), (DA). But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, i.e. (AB) = (CD) and (BC) = (DA), the law can be stated as : 2(AB)^2+2(BC)^2=(AC)^2+(BD)^2\, If the parallelogram is a rectangle, the two diagonals are of equal lengths (AC) = (BD), so : 2(AB)^2+2(BC)^2=2(AC)^2\, and the statement reduces to the Pythagorean theorem. For the general quadrilateral with four sides not necessarily equal, : (AB)^2+(BC)^2+(CD)^2+(DA)^2=(AC)^2+(BD)^2+4x^2, where x is the length of the line segment joining the midpoints of the diagonals.
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These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors. The envelope of the infinitude of area bisectors is a deltoid (broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set). The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the extended sides of the triangle.
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Two chandeliers were suspended at opposite ends of this central oval, which was unlike most other Parisian theatres, where typically a single chandelier hung from the center of the ceiling and sometimes obstructed views of the stage from the galleries. Surrounding the scene with Apollo were painted in perspective a balustrade topped by a colonnade of double Corinthian columns. The colonnade was interrupted at the midpoints between the vertices by four thrones occupied by the muses of Painting, Comedy, Music, and Tragedy. The theatre was designed to accommodate two divergent types of audience, that of the working class common to the boulevard du Temple and that of the most brilliant society of Paris, on whom the directors of the theatre depended as their patrons.
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Developed by Allied Properties Real Estate Investment Trust and designed by Sweeny&Co; Architects Inc. The concept aims to preserve the existing brick buildings adjacent to each other while adding a new 12-story office tower perched at the top creating the iconic 24m tall L-shaped atrium. In order to achieve this feat, the team designed a massive table top structure supported by three giant “delta frame” steel columns which came down through the atrium with an additional of 8 concrete columns hidden within one of the existing brick buildings. The delta frame, which consists of four angled tubular columns connected at the midpoints, was an innovative design approach which provided the lateral stability needed to support the large complex structure.
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To see that the incenter of the medial triangle coincides with the intersection point of the cleavers, consider a homogeneous wireframe in the shape of triangle ABC consisting of three wires in the form of line segments having lengths a, b, c. The wire frame has the same center of mass as a system of three particles of masses a, b, c placed at the midpoints D, E, F of the sides BC, CA, AB. The centre of mass of the particles at E and F is the point P which divides the segment EF in the ratio c : b. The line DP is the internal bisector of ∠D. The centre of mass of the three particle system thus lies on the internal bisector of ∠D.
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When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median m is 3 since the median is the smallest value of x for which F(x) is greater than a half.
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In any triangle all of the following nine points are concyclic on what is called the nine-point circle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter are concyclic. If lines are drawn through the Lemoine point parallel to the sides of a triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, in what is called the Lemoine circle. The van Lamoen circle associated with any given triangle T contains the circumcenters of the six triangles that are defined inside T by its three medians.
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The different layout was primarily chosen to reduce the impact of the inertial torsion inherent with crank throws spaced 90° apart due to the pistons being accelerated (start-stop motion), given this engine was meant to be high revving and inertial forces scale as to the square of engine speed. The reduction in torsion was achieved by splitting the crank into two separate parts, geared together, from their respective midpoints, via a counter-shaft, from which power was delivered to the gearbox.Entwicklungsgeschichte des URS-Rennmotor, German language article on the developmental history of the URS engine. It is likely this inertial torsion within the crank is the reason for Yamaha citing crank forging improvements as a reason for the cross-plane crank being viable in a road bike.
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For the set {1, 2, 4, 5, 10, 11, 13, 14}, all midpoints of two elements (the 28 yellow points) land outside the set, so no three elements can form an arithmetic progression In mathematics, and in particular in arithmetic combinatorics, a Salem-Spencer set is a set of numbers no three of which form an arithmetic progression. Salem–Spencer sets are also called 3-AP-free sequences or progression-free sets. They have also been called non-averaging sets, but this term has also been used to denote a set of integers none of which can be obtained as the average of any subset of the other numbers. Salem-Spencer sets are named after Raphaël Salem and Donald C. Spencer, who showed in 1942 that Salem–Spencer sets can have nearly-linear size.
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The Bricard octahedra all have an axis of 180° rotational symmetry, and are formed from any three pairs of points such that each pair is symmetric around the same axis and there is no plane containing all six points. (For instance, the six points of a regular octahedron can be paired up in this way by an axial symmetry around a line through two opposite edge midpoints, although the Bricard octahedron resulting from this pairing would not be regular.) The octahedra have 12 edges, each of which connects two points that do not belong to the same symmetric pair as each other. These edges form the octahedral graph . Each of the eight triangular faces of these octahedra connects three points, one from each symmetric pair, in all of the eight possible ways of doing this...
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A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of orthocentric tetrahedron. An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex. A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median and a line segment joining the midpoints of two opposite edges is called a bimedian of the tetrahedron.
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An alternative construction, the medial graph, coincides with the line graph for planar graphs with maximum degree three, but is always planar. It has the same vertices as the line graph, but potentially fewer edges: two vertices of the medial graph are adjacent if and only if the corresponding two edges are consecutive on some face of the planar embedding. The medial graph of the dual graph of a plane graph is the same as the medial graph of the original plane graph.. For regular polyhedra or simple polyhedra, the medial graph operation can be represented geometrically by the operation of cutting off each vertex of the polyhedron by a plane through the midpoints of all its incident edges.. This operation is known variously as the second truncation,. degenerate truncation,.
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A triangle showing its circumcircle and circumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue) In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices. The nine-point center is listed as point X(5) in Clark Kimberling's Encyclopedia of Triangle Centers..Encyclopedia of Triangle Centers, accessed 2014-10-23.
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Now the midpoints of the sides of any triangle are the images of its vertices by a homothety with factor −½, centered at the barycenter of the triangle. Applied to the anticomplementary triangle, which is itself obtained from the Johnson triangle by a homothety with factor 2, it follows from composition of homotheties that the reference triangle is homothetic to the Johnson triangle by a factor −1. Since such a homothety is a congruence, this gives property 5, and also the Johnson circles theorem since congruent triangles have circumscribed circles of equal radius. For property 6, it was already established that the perpendicular bisectors of the sides of the anticomplementary triangle all pass through the point H; since that side is parallel to a side of the reference triangle, these perpendicular bisectors are also the altitudes of the reference triangle.
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Fort Wood's star-shaped walls became the base of the Statue of Liberty. Prisms over the hendecagrams {11/3} and {11/4} may be used to approximate the shape of DNA molecules. An 11-pointed star from the Momine Khatun Mausoleum Fort Wood, now the base of the Statue of Liberty in New York City, is a star fort in the form of an irregular 11-point star.. The Topkapı Scroll contains images of an 11-pointed star Girih form used in Islamic art. The star in this scroll is not one of the regular forms of the hendecagram, but instead uses lines that connect the vertices of a hendecagon to nearly-opposite midpoints of the hendecagon's edges.. 11-pointed star Girih patterns are also used on the exterior of the Momine Khatun Mausoleum; Eric Broug writes that its pattern "can be considered a high point in Islamic geometric design".
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In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as for making a histogram, effectively replacing the values by the midpoints of the intervals they are assigned to. The mode is then the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable. An alternate approach is kernel density estimation, which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.
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The circle is an instance of a conic section and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle ABC and a fourth point P, where the particular nine-point circle instance arises when P is the orthocenter of ABC. The vertices of the triangle and P determine a complete quadrilateral and three "diagonal points" where opposite sides of the quadrilateral intersect. There are six "sidelines" in the quadrilateral; the nine-point conic intersects the midpoints of these and also includes the diagonal points. The conic is an ellipse when P is interior to ABC or in a region sharing vertical angles with the triangle, but a nine-point hyperbola occurs when P is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of ABC.
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To illustrate the potential of the k-means algorithm to perform arbitrarily poorly with respect to the objective function of minimizing the sum of squared distances of cluster points to the centroid of their assigned clusters, consider the example of four points in R2 that form an axis-aligned rectangle whose width is greater than its height. If k = 2 and the two initial cluster centers lie at the midpoints of the top and bottom line segments of the rectangle formed by the four data points, the k-means algorithm converges immediately, without moving these cluster centers. Consequently, the two bottom data points are clustered together and the two data points forming the top of the rectangle are clustered together--a suboptimal clustering because the width of the rectangle is greater than its height. Now, consider stretching the rectangle horizontally to an arbitrary width.
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The antiparallelogram is an important feature in the design of Hart's inversor, a linkage that (like the Peaucellier–Lipkin linkage) can convert rotary motion to straight-line motion.. An antiparallelogram-shaped linkage can also be used to connect the two axles of a four-wheeled vehicle, decreasing the turning radius of the vehicle relative to a suspension that only allows one axle to turn.. A pair of nested antiparallelograms was used in a linkage defined by Alfred Kempe as part of his universality theorem stating that any algebraic curve may be traced out by the joints of a suitably defined linkage. Kempe called the nested-antiparallelogram linkage a "multiplicator", as it could be used to multiply an angle by an integer. Antiparallelogram braced to stop it turning into a normal parallelogram. The points PQRS are the midpoints of the sides and are collinear, and X can be any distance away on the perpendicular bisector of SQ. With this linkage .
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A pay scale (also known as a salary structure) is a system that determines how much an employee is to be paid as a wage or salary, based on one or more factors such as the employee's level, rank or status within the employer's organization, the length of time that the employee has been employed, and the difficulty of the specific work performed. Examples of pay scales include U.S. uniformed services pay grades, the salary grades by which United States military personnel are paid, and the General Schedule, the salary grades by which United States white-collar civil service personnel are paid. Private employers use salary structures with grades (including minimums, midpoints and maximums) to define the ranges of pay available to employees in each grade/range. In 2002, Gregory A. Stoskopf identified a trend of employers to align their salary structures to the competitive values of jobs in the appropriate labor market.
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These are paths on the surface of the polyhedron that avoid the vertices and locally look like a shortest path: they follow straight line segments across each face of the polyhedron that they intersect, and when they cross an edge of the polyhedron they make complementary angles on the two incident faces to the edge. Intuitively, one could stretch a rubber band around the polyhedron along this path and it would stay in place: there is no way to locally change the path and make it shorter. For example, one type of geodesic crosses the two opposite edges of the snub disphenoid at their midpoints (where the symmetry axis exits the polytope) at an angle of /3. A second type of geodesic passes near the intersection of the snub disphenoid with the plane that perpendicularly bisects the symmetry axis (the equator of the polyhedron), crossing the edges of eight triangles at angles that alternate between /2 and /6.
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A tetrahedron is a disphenoid if and only if its circumscribed parallelepiped is right-angled. We also have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide.. Another characterization states that if d1, d2 and d3 are the common perpendiculars of AB and CD; AC and BD; and AD and BC respectively in a tetrahedron ABCD, then the tetrahedron is a disphenoid if and only if d1, d2 and d3 are pairwise perpendicular.. The disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed geodesics are non-self-intersecting.. The disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area, and the tetrahedra in which the angular defects of all four vertices equal . They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints..
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If M1 and M2 are the midpoints of the diagonals AC and BD respectively in a tangential quadrilateral ABCD with incenter I, and if the pairs of opposite sides meet at J and K with M3 being the midpoint of JK, then the points M3, M1, I, and M2 are collinear. The line containing them is the Newton line of the quadrilateral. If the extensions of opposite sides in a tangential quadrilateral intersect at J and K, and the extensions of opposite sides in its contact quadrilateral intersect at L and M, then the four points J, L, K and M are collinear.. If the incircle is tangent to the sides AB, BC, CD, DA at T1, T2, T3, T4 respectively, and if N1, N2, N3, N4 are the isotomic conjugates of these points with respect to the corresponding sides (that is, AT1 = BN1 and so on), then the Nagel point of the tangential quadrilateral is defined as the intersection of the lines N1N3 and N2N4. Both of these lines divide the perimeter of the quadrilateral into two equal parts.
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The snub disphenoid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well- covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal bipyramid, and an irregular polyhedron with 12 vertices and 20 triangular faces.. The snub disphenoid has the same symmetries as a tetragonal disphenoid: it has an axis of 180° rotational symmetry through the midpoints of its two opposite edges, two perpendicular planes of reflection symmetry through this axis, and four additional symmetry operations given by a reflection perpendicular to the axis followed by a quarter-turn and possibly another reflection parallel to the axis.. That is, it has antiprismatic symmetry, a symmetry group of order 8\. Spheres centered at the vertices of the snub disphenoid form a cluster that, according to numerical experiments, has the minimum possible Lennard-Jones potential among all eight-sphere clusters.. Up to symmetries and parallel translation, the snub disphenoid has five types of simple (non-self-crossing) closed geodesics.
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