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In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle. This circle is called the circumcircle or circumscribed circle , and the vertices are said to be concyclic .
[15] [17] The right kites are exactly the kites that are cyclic quadrilaterals, meaning that there is a circle that passes through all their vertices. [18] 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 ...
More generally, a polygon in which all vertices are concyclic is called a cyclic polygon. A polygon is cyclic if and only if the perpendicular bisectors of its edges are concurrent. [10] Every regular polygon is a cyclic polygon. For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular.
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.
In Euclidean geometry, a right kite is a kite (a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle. [1] That is, it is a kite with a circumcircle (i.e., a cyclic kite). Thus the right kite is a convex quadrilateral and has two opposite right ...
In Euclidean geometry, a harmonic quadrilateral, or harmonic quadrangle, [1] is a quadrilateral that can be inscribed in a circle (cyclic quadrilateral) in which the products of the lengths of opposite sides are equal. It has several important properties.
The quadrilateral case follows from a simple extension of the Japanese theorem for cyclic quadrilaterals, which shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry. [2]
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]