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Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, [13] which by Brahmagupta's formula all have the same area. Specifically, for sides a , b , c , and d , side a could be opposite any of side b , side c , or side d .
Ptolemy's theorem is a relation among these lengths in a cyclic quadrilateral. = + In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle).
The theorem states that for a quadrilateral inscribed in a circle, the product of the lengths of the diagonals equals the sum of the products of the two pairs of lengths of opposite sides. The derivations of trigonometric identities rely on a cyclic quadrilateral in which one side is a diameter of the circle.
Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero. Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths , , , as = () () where = (+ + +) is the semiperimeter. Heron's formula is ...
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
A convex quadrilateral is cyclic if and only if opposite angles sum to 180°. Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral. Harmonic quadrilateral: a cyclic quadrilateral such that the products of the lengths of the opposing sides are equal. Bicentric quadrilateral: it is both tangential and cyclic.
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 lengths of the diagonals in a bicentric quadrilateral can be expressed in terms of the sides or the tangent lengths, which are formulas that holds in a cyclic quadrilateral and a tangential quadrilateral respectively. In a bicentric quadrilateral with diagonals p, q, the following identity holds: [11]