Kitap 3
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Again, since the point G is the centre of the circle ADE, GA is equal to GD.
12
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.
12
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.
12
Therefore etc. Q. E. D.] will not fail to pass. The Greek has the double negative, οὐκ ἄρα ἡ...εὐθεῖα... οὐκ ἐλεύσεται, literally the straight line...will not not-pass....
PROPOSITION 13.
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A circle does not touch a circle at more points than one, whether it touch it internally or externally.
13
For, if possible, let the circle ABDC touch the circle EBFD, first internally, at more points than one, namely D, B.
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Let the centre G of the circle ABDC, and the centre H of EBFD, be taken.
13
Therefore the straight line joined from G to H will fall on B, D. [III. 11]
13
Let it so fall, as BGHD.
13
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.
13
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.
13
Therefore a circle does not touch a circle internally at more points than one.
13
I say further that neither does it so touch it externally.
