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. * For let two circles, $ABC$ and $ADE$, touch one another internally at point $A$, and let the center $F$ of circle $ABC$ have been found [Prop. 3.1], and (the center) $G$ of (circle) $ADE$ [Prop. 3.1]. * I say that the straight line joining $G$ to $F$, being produced, will fall on $A$.


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