Proposition 13

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.

Therefore the straight line joined from G to H falls on B and D.

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.

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.