---
book: 3
number: 6
id: "III.6"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.6"
license: "CC-BY-SA-4.0"
---

# III.6

*If two circles touch one another*, *they will not have the same centre*.

## Proof

For let the two circles *ABC*, *CDE* touch one another at the point *C*; I say that they will not have the same centre.

For, if possible, let it be *F*; let *FC* be joined, and let *FEB* be drawn through at random.

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

Again, since the point *F* is the centre of the circle *CDE*, *FC* is equal to *FE*.

But *FC* was proved equal to *FB*; therefore *FE* is also equal to *FB*, the less to the greater: which is impossible.

Therefore *F* is not the centre of the circles *ABC*, *CDE*.

Therefore etc. Q. E. D.