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Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
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.
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 .
Equidiagonal quadrilateral: the diagonals are of equal length. Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram. Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
There are 2 dihedral subgroups: Dih 2, Dih 1, and 3 cyclic subgroups: Z 4, Z 2, and Z 1. A square is a special case of many lower symmetry quadrilaterals: A rectangle with two adjacent equal sides; A quadrilateral with four equal sides and four right angles; A parallelogram with one right angle and two adjacent equal sides; A rhombus with a ...
A rhombus has an inscribed circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.
[2] [3] A kite may also be called a dart, [4] particularly if it is not convex. [5] [6] Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and ...
The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length , while AC (shown in red) is a face diagonal and has length . In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.