and therefore the centre of each circle is in the straight line GH which bisects BD at right angles; [III. 1, Corol.
therefore GH passes through the point of contact. [III. 11.
But GH does not pass through the point of contact, because the points B, D are out of the line GH;
which is absurd.
Therefore one circle cannot touch another on the inside at more points than one.
Nor can one circle touch another on the outside at more points than one.
For, if it be possible, let the circle ACK touch the circle ABG at the points A, C. Join AC.
Then, because the two points A, C are in the circumference of the circle ACK, the straight line AC which joins them, falls within the circle ACK; [III. 2.
but the circle ACK is without the circle ABC; [Hypothesis.
therefore the straight line AC is without the circle ABC.
But because the two points A,C are in the circumference of the circle ABC, the straight line AC falls within the circle ABC; [III. 2.
which is absurd.
Therefore one circle cannot touch another on the outside at more points than one.
And it has been shewn that one circle cannot touch another on the inside at more points than one.
Wherefore, one circle &c. q.e.d.
