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A root-phi rectangle divides into a pair of Kepler triangles (right triangles with edge lengths in geometric progression). The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles.
The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals p and q: [7] K = p q 2 . {\displaystyle K={\frac {pq}{2}}.} Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal. [ 5 ]
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e and f to give [2] [3]
The diagonal formula can refer to: some geometric method see Polygon entry The formula developed by Aumann and Shapley to construct a Shapley value for non atomic games with a continuum of players
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem .
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
Get ready for all of today's NYT 'Connections’ hints and answers for #551 on Friday, December 13, 2024. Today's NYT Connections puzzle for Friday, December 13, 2024 The New York Times
This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral—when A + C = 180°. Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is = + .
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