Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/108

84 PROPOSITION 12. THEOREM.

If two circles touch one another externally, the straight line which joins their centres shall pass through the point of contact.

Let the two circles ABC, ADE touch one another externally at the point A; and let F be the centre of the circle ABC, and G the centre of the circle ADE: the straight line which joins the points F, G, shall pass through the point A.

For, if not, let it pass otherwise, if possible, as FCDG, and join FA, AG.

Then, because F is the centre of the circle ABC, FA is equal to FC; [I.Def.16. and because G is the centre of the circle ADE, GA is equal to GD; therefore FA, AG are equal to FC, DG. [Axiom 2. Therefore the whole FG is greater than FA, AG. But FG is also less than FA, AG; [I. 20. which is impossible. Therefore the straight line which joins the points F, G, cannot pass otherwise than through the point A, that is, it must pass through A.

Wherefore, if two circles &c.

PROPOSITION 13. THEOREM.

One circle cannot touch another at more points than one, whether it touches it on the inside or outside.

For, if it be possible, let the circle EBF touch the circle ABC at more points than one; and first on the inside, at the points B, D. Join BD, and draw GH bisecting BD at right angles. [I. 10, 11. Then, because the two points B, D are in the circumference of each of the circles, the straight line BD falls within each of them; [III. 2.