---
book: 3
number: 11
id: "III.11"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.11"
license: "CC-BY-SA-4.0"
---

# III.11

*If two circles touch one another internally*, *and their centres be taken*, *the straight line joining their centres*, *if it be also produced*, *will fall on the point of contact of the circles*.

## Proof

For let the two circles *ABC*, *ADE* touch one another internally at the point *A*, and let the centre *F* of the circle *ABC*, and the centre *G* of *ADE*, be taken; I say that the straight line joined from *G* to *F* and produced will fall on *A*.

For suppose it does not, but, if possible, let it fall as *FGH*, and let *AF*, *AG* be joined.

Then, since *AG*, *GF* are greater than *FA*, that is, than *FH*,

let *FG* be subtracted from each; therefore the remainder *AG* is greater than the remainder *GH*.

But *AG* is equal to *GD*; therefore *GD* is also greater than *GH*, the less than the greater: which is impossible.

Therefore the straight line joined from *F* to *G* will not fall outside; therefore it will fall at *A* on the point of contact.

Therefore etc. Q. E. D.