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Examples of cyclic quadrilaterals. 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.
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).
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]
In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
[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 ...
In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect. [3] Brahmagupta's theorem states that for a cyclic orthodiagonal quadrilateral, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side. [3]
Then the quadrilateral formed by M 1, M 2, M 3, M 4 is a rectangle. Proofs are given by Bogomolny [2] and Reyes. [1] This theorem may be extended to prove the Japanese theorem for cyclic polygons, according to which the sum of inradii of a triangulated cyclic polygon does not depend on how it is triangulated. The special case of the theorem for ...
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.