Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally

(Proposition 11 from Book 3 of Euclid's “Elements”)

If two circles touch one another internally, and their centers are found, then the straight line joining their centers, being produced, will fall upon the point of union of the circles.


Modern Formulation

If two circles touch one another internally at a point $A$, then the straight line going through their centers goes also through $A$.

Proofs: 1

Proofs: 1 2
Sections: 3

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