Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

---

---

I say that the straight line joined from F to G will pass through the point of contact at A.

For suppose it does not,
but, if possible, let it pass as FCDG, and let AF, AG be joined.

Then, since the point F is the centre of the circle ABC,

FA is equal to FC.

Again, since the point G is the centre of the circle ADE,

GA is equal to GD.

But FA was also proved equal to FC;

therefore FA, AG are equal to FC, GD, so that the whole FG is greater than FA, AG; but it is also less [I. 20]: which is impossible.

Therefore the straight line joined from F to G will not fail to pass through the point of contact at A;

therefore it will pass through it.

Therefore etc. Q. E. D.] ¹

---

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 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;