A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle, meaning that there exists a circle that passes through all four vertices of the quadrilateral.
Cyclic quadrilaterals are useful in various types of geometry problems, particularly those in which angle chasing is required. It is not unusual, for instance, to intentionally add points (and lines) to diagrams in order to exploit the properties of cyclic quadrilaterals.
which leads to the following two results:
The opposite angles of a cyclic quadrilateral add to , or radians.
In a cyclic quadrilateral , we have
and similar relations e.g.
These can both be directly verified from the above angle equalities.
Also recall that , where is the center of the circle, by the inscribed angle theorem. This can also lead to useful information, if the center of the circumcircle is relevant.
The sides and diagonals of a cyclic quadrilateral are closely related:
In cyclic quadrilateral ,
In other words, the product of the lengths of the diagonals is equal to the sum of the products of opposite sides.
In fact, it is true of any quadrilateral that
meaning that the cyclic quadrilateral is the equality case of this inequality.
In fact, more can be said about the diagonals: if are the lengths of the sides of the quadrilateral (in clockwise order),
which also demonstrates Ptolemy's theorem.
The cyclic quadrilateral is the equality case of another inequality: given four side lengths, the cyclic quadrilateral maximizes the resulting area.
Let a cyclic quadrilateral have side lengths , and let be called the semiperimeter. Then the area of the quadrilateral is equal to
The area is then given by a special case of Bretschneider's formula.
Here are few well-known problems which use the basic properties of cyclic quadrilaterals. They are mainly of Olympiad flavor and are solvable by elementary methods.
Problem 1. Let and be two points on side and of square , such that . Let and be the intersection of diagonal with and respectively. Let be the intersection of and . Prove that is perpendicular to .
Problem 2. is inscribed in the circle centered at such that the angles and are acute. If is its orthocenter, then prove that .
Problem 3. Let and be the feet of the altitudes of Prove that the altitudes of are the angle bisectors of