GeoProof — Cyclic Quadrilateral explorer

Cyclic Quadrilateral

A cyclic quadrilateral has all four vertices on one circle. Its opposite angles are always supplementary:

∠A + ∠C = 180° and ∠B + ∠D = 180°

Why it's true

An inscribed angle equals half the arc it opens onto. ∠A and ∠C open onto the two arcs that together make the whole circle (360°), so their halves add to 180°. The same argument gives ∠B + ∠D = 180°.

The converse

It also runs the other way: if a quadrilateral's opposite angles sum to 180°, its four corners must lie on a common circle. That's a standard test for whether four points are concyclic.

Related results

The same four concyclic points also satisfy Ptolemy's theorem (a product relation between the sides and diagonals) and, for cyclic quadrilaterals, Brahmagupta's area formula — both explored elsewhere in GeoProof.