| A circle does not touch another circle at more than one point whether it touches it internally or externally. | ||
| For, if possible, let the circle ABDC touch the circle EBFD, first internally, at more points than one, namely D and B. | ||
| Take the center G of the circle ABDC and the center H of EBFD. | III.1 | |
| Therefore the straight line joined from G to H falls on B and D. | III.11 | |
| Let it so fall, as BGHD.
Then, since the point G is the center of the circle ABCD and BG equals GD, therefore BG is greater than HD. Therefore BH is much greater than HD. | ||
|
Again, since the point H is the center of the circle EBFD, BH equals 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 and C. Join AC. | ||
| Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd. | III.2 | |
| 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 a circle does not touch another circle at more than one point whether it touches it internally or externally. | ||
| Q.E.D. | ||
There are logical flaws in this proof similar to those in the last two proofs.
This proposition is not used in the rest of the Elements.
Book III Introduction - Proposition III.12 - Proposition III.14.