---
book: 3
number: 13
id: "III.13"
kind: "theorem"
uses: ["[[book-3/proposition-11]]", "[[book-3/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.13"
license: "CC-BY-SA-4.0"
---

# III.13

*A circle does not touch a circle at more points than one*, *whether it touch it internally or externally*.

## Proof

For, if possible, let the circle *ABDC* touch the circle *EBFD*, first internally, at more points than one, namely *D*, *B*.

Let the centre *G* of the circle *ABDC*, and the centre *H* of *EBFD*, be taken.

Therefore the straight line joined from *G* to *H* will fall on *B*, *D*. [[book-3/proposition-11|III. 11]]

Let it so fall, as *BGHD*.

Then, since the point *G* is the centre of the circle *ABCD*, *BG* is equal to *GD*; therefore *BG* is greater than *HD*; therefore *BH* is much greater than *HD*.

Again, since the point *H* is the centre of the circle *EBFD*, *BH* is equal to *HD*; but it was also proved much greater than it: which is impossible.

Therefore a circle does not touch a circle internally at more points than one.

I say further that neither does it so touch it externally.

For, if possible, let the circle *ACK* touch the circle *ABDC* at more points than one, namely *A*, *C*, and let *AC* be joined.

Then, since on the circumference of each of the circles *ABDC*, *ACK* two points *A*, *C* have been taken at random, the straight line joining the points will fall within each circle; [[book-3/proposition-2|III. 2]] but it fell within the circle *ABCD* and outside *ACK* [[book-3/definitions#Definition 3|III. Def. 3]]: which is absurd.

Therefore a circle does not touch a circle externally at more points than one.

And it was proved that neither does it so touch it internally.

Therefore etc. Q. E. D.