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 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. ...
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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