Proposition: 3.13: Circles Touch at One Point at Most

(Proposition 13 from Book 3 of Euclid's “Elements”)

A circle does not touch a(nother) circle at more than one point, whether they touch internally or externally.

Modern Formulation

Two circles touch one another (internally or externally) at exactly one point.

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