[Folio 0038 | p. 32]

PROPOSITION 13.

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

For, if possible, let the circle ABDC[1] 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. [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;

[Folio 0039 | p. 33]

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[2] externally.

For, if possible, let the circle ACK touch the circle ABDC[1] 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; [III. 2] but it fell within the circle ABCD and outside ACK [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.

[1] (1,2)

ABDC. Euclid writes ABCD (here and in the next proposition), notwithstanding the order in which the points are placed in the figure.

[2]

does it so touch it. It is necessary to supply these words which the Greek ($\text{ὅτι οὐδὲ ἐκτός}$ and $\text{ὅτι οὐδὲ ἐντός}$) leaves to be understood.

Dependency Graph

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See Commentary on Proposition III.13.