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;
therefore BG is greater than HD;
therefore BH is much greater than HD.
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.
Therefore a circle does not touch a circle internally at more points than one.