Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral

(Proposition 22 from Book 3 of Euclid's “Elements”)

For quadrilaterals within circles, the (sum of the) opposite angles is equal to two right angles.

Modern Formulation

In a quadrilateral built upon four secants of a given circle, the sum of the opposite angles is $180^\circ$.

Proofs: 1

Proofs: 1 2

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016