In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. The vertices are said to be concyclic. Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... Other uses: Quadrilateral (disambiguation) In geometry, a quadrilateral is a polygon with four sides and four vertices. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ... In geometry, a set of points is said to be concyclic if they lie on a common circle. ...
Opposite angles are supplementary angles (adding up to either 180 in degrees or π in radians).
The area of a cyclic quadrilateral is maximal among all quadrilaterals having the same side lengths.
Exterior angles are equal to the opposite interior angles.
When the diagonals are drawn, two pairs of similar triangles are formed.
The product of the two diagonals is equal to the sum of the products of opposite sides. (Ptolemy's Theorem)
See also:cyclic polygon. In geometry, Brahmaguptas formula formula finds the area of any quadrilateral. ... In geometry, Herons formula (also called Heros formula) states that the area of a triangle whose sides have lengths a, b and c is where s is the triangles semiperimeter: (see also square root). ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... Several equivalence relations in mathematics are called similarity. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...
External links
Cyclic quadrilateral theorem by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas".
The length of the two diagonals of a cyclicquadrilateral are related to the four sides in Ptolemy's Theorem which states (using m and n for the diagonals lengths) mn=ac+bd.
The center of this "orthic cyclicquadrilateral" is the reflection of the circumcenter of the original quadrilateral in the anti-center.
The anti-center of the orthic quadrilateral is the same as the anti-center of the original quadrilateral, and so the orthocenters of the triangles formed by the orthic quadrilateral are the vertices of the original cyclicquadrilateral.
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