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Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally

(Proposition 12 from Book 3 of Euclid's “Elements”)

If two circles touch one another externally then the (straight line) joining their centers will go through the point of union.

• For let two circles, $ABC$ and $ADE$, touch one another externally at point $A$, and let the center $F$ of $ABC$ have been found [Prop. 3.1], and (the center) $G$ of $ADE$ [Prop. 3.1].
• I say that the straight line joining $F$ to $G$ will go through the point of union at $A$.

Modern Formulation

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

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References

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", https://proofwiki.org/wiki/Main_Page, 2016