Cyclic Quadrilateral - Other Properties

Other Properties

• In a cyclic quadrilateral ABCD, the incenters in triangles ABC, BCD, CDA, and DAB are the vertices of a rectangle. This is one of the theorems known as the Japanese theorem. The orthocenters of the same four triangles are the vertices of a quadrilateral congruent to ABCD, and the centroids in those four triangles are vertices of another cyclic quadrilateral.

• In a cyclic quadrilateral ABCD with circumcenter O, let P be the point where the diagonals AC and BD intersect. Then angle APB is the arithmetic mean of the angles AOB and COD. This is a direct consequence of the inscribed angle theorem and the exterior angle theorem.

• There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression.

• If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.

• If the opposite sides of a cyclic quadrilateral are extended to meet at E and F, then the internal angle bisectors of the angles at E and F are perpendicular.