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Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side. [23] If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side. [23]
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.
In geometry, 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 bisects the opposite side. [1] It is named after the Indian mathematician Brahmagupta (598-668). [2]
[15] [16] The right kites are exactly the kites that are cyclic quadrilaterals, meaning that there is a circle that passes through all their vertices. [17] The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles are supplementary (they add to 180°); if one pair is supplementary the other is as well ...
Ptolemy's inequality is often stated for a special case, in which the four points are the vertices of a convex quadrilateral, given in cyclic order. [2] [3] However, the theorem applies more generally to any four points; it is not required that the quadrilateral they form be convex, simple, or even planar.
If an orthodiagonal quadrilateral is also cyclic, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the length of the opposite side. [2] In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where ...
The two bimedians of a quadrilateral (segments joining midpoints of opposite sides) and the line segment joining the midpoints of the diagonals are concurrent and are all bisected by their point of intersection. [3]: p.125 In a tangential quadrilateral, the four angle bisectors concur at the center of the incircle. [4]