Book 3
11
Therefore the straight line joined from F to G will not fall outside; therefore it will fall at A on the point of contact.
11
Therefore etc. Q. E. D. the straight line joining their centres, literally the straight line joined to their centres (ἡ ἐπὶ τὰ κέντρα αὐτῶν ἐπιζευγνυμένη εὐθεῖα). point of contact is here συναφἡ, and in the enunciation of the next proposition ἐπαφἡ.
PROPOSITION 12.
12
If two circles touch one another externally, the straight line joining their centres will pass through the point of contact.
12
For let the two circles ABC, ADE touch one another externally at the point A, and let the centre F of ABC, and the centre G of ADE, be taken; I say that the straight line joined from F to G will pass through the point of contact at A.
12
For suppose it does not, but, if possible, let it pass as FCDG, and let AF, AG be joined.
12
Then, since the point F is the centre of the circle ABC, FA is equal to FC.
12
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.
13
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.
13
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.
