Proposition: 3.10: Two Circles have at most Two Points of Intersection

(Proposition 10 from Book 3 of Euclid's “Elements”)

A circle does not cut a(nother) circle at more than two points.

Modern Formulation

Two circles cut or touch one another in at most two points.

Proofs: 1

Proofs: 1

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