---
book: 3
number: 5
id: "III.5"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.5"
license: "CC-BY-SA-4.0"
---

# III.5

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

## Proof

For let the circles *ABC*, *CDG* cut one another at the points *B*, *C*; I say that they will not have the same centre.

For, if possible, let it be *E*; let *EC* be joined, and let *EFG* be drawn through at random.

Then, since the point *E* is the centre of the circle *ABC*, *EC* is equal to *EF*. [[book-1/definitions#Definition 15|I. Def. 15]]

Again, since the point *E* is the centre of the circle *CDG*, *EC* is equal to *EG*.

But *EC* was proved equal to *EF* also; therefore *EF* is also equal to *EG*, the less to the greater : which is impossible.

Therefore the point *E* is not the centre of the circles *ABC*, *CDG*.

Therefore etc. Q. E. D.