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.



In a quadrilateral $ABCD$:

  • $\angle A + \angle C = \angle B + \angle D = {180}^{o}$ This property is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
  • $\angle ABD = \angle ACD$
  • $\angle BCA = \angle BDA$
  • $\angle BAC = \angle BDC$
  • $\angle CAD = \angle CBD$
  • 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:

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