Cyclic Quadrilaterals

A quadrilateral is cyclic if its four vertices lie on a single circle. These are among the most useful configurations in olympiad geometry because they link angles and lengths powerfully.

Identifying Cyclic Quads

A quadrilateral $A BC D$ is cyclic if and only if any one of these conditions holds:

[cite_start]Opposite Angles: $∠ A + ∠ C = 18 0^{\circ}$ . [cite: 68]
Exterior Angle: The exterior angle at one vertex equals the interior opposite angle.
Angles Subtended by Side: $∠ D A C = ∠ D BC$ . (The "Bowtie" or "Butterfly" property).
Power of a Point: $P A \cdot PC = PB \cdot P D$ (where $P$ is intersection of diagonals).

[Image of Cyclic quadrilateral properties]

Ptolemy's Theorem

Theorem (Ptolemy's Theorem)

For a cyclic quadrilateral

A BC D

with diagonals

A C

and

B D

A B \cdot C D + BC \cdot D A = A C \cdot B D

"The product of the diagonals equals the sum of the products of opposite sides."

Ptolemy's Inequality: For any quadrilateral, $A B \cdot C D + BC \cdot D A \geq A C \cdot B D$ , with equality iff it is cyclic.

Practice Problems

Exercise (Problem 1)

Let

H

be the orthocenter of acute

△ A BC

. Prove that the quadrilaterals

A B D E

(where

D, E

are feet of altitudes) and

C DH E

are cyclic.

Exercise (Problem 2)

Use Ptolemy's Theorem on a regular pentagon to prove that the ratio of the diagonal to the side is the golden ratio

ϕ = \frac{1 + 5}{2}

Exercise (Problem 3)

Point

P

lies on the circumcircle of equilateral triangle

A BC

. Prove that the distance from

P

to the furthest vertex is the sum of the distances to the other two vertices.