Proof: By Euclid
(related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
- For (if) not then, if possible, let it fall like $FGH$ (in the figure), and let $AF$ and $AG$ have been joined.
- Therefore, since $AG$ and $GF$ is greater than $FA$, that is to say $FH$ [Prop. 1.20], let $FG$ have been taken from both.
- Thus, the remainder $AG$ is greater than the remainder $GH$.
- And $AG$ (is) equal to $GD$.
- Thus, $GD$ is also greater than $GH$, the lesser than the greater.
- The very thing is impossible.
- Thus, the straight line joining $F$ to $G$ will not fall outside (one circle but inside the other).
- Thus, it will fall upon the point of union (of the circles) at point $A$.
- Thus, 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.
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"