Cyclic quadrilateral

A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. While all triangles are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties.

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Properties

In a quadrilateral :

• This property is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
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• All four perpendicular bisectors are concurrent. The converse is also true. This intersection is the circumcenter of the quadrilateral.
• Any two opposite sites of the quadrilateral are antiparallel with respect to the other two opposite sites.

Applicable Theorems/Formulae

The following theorems and formulae apply to cyclic quadrilaterals:

• Ptolemy's Theorem
• Brahmagupta's formula

Problems

Intermediate/Advanced

• 1991 AIME Problems/Problem 12
• 2001 AIME I Problems/Problem 13
• 2000 AIME I Problems/Problem 14
• 1997 AIME Problems/Problem 15

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