---
book: 3
number: 12
id: "III.12"
kind: "theorem"
uses: ["[[book-1/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.12"
license: "CC-BY-SA-4.0"
---

# III.12

*If two circles touch one another externally*, *the straight line joining their centres will pass through the point of contact*.

## Proof

For let the two circles *ABC*, *ADE* touch one another externally at the point *A*, and let the centre *F* of *ABC*, and the centre *G* of *ADE*, be taken; I say that the straight line joined from *F* to *G* will pass through the point of contact at *A*.

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

Then, since the point *F* is the centre of the circle *ABC*, *FA* is equal to *FC*.

Again, since the point *G* is the centre of the circle *ADE*, *GA* is equal to *GD*.

But *FA* was also proved equal to *FC*; therefore *FA*, *AG* are equal to *FC*, *GD*, so that the whole *FG* is greater than *FA*, *AG*; but it is also less [[book-1/proposition-20|I. 20]]: which is impossible.

Therefore the straight line joined from *F* to *G* will not fail to pass through the point of contact at *A*; therefore it will pass through it.

Therefore etc. Q. E. D.]