A circle does not cut a circle at more points than two.
For, if possible, let the circle ABC cut the circle DEF at more points than
two, namely B, C, F, H;
let BH, BG be joined and bisected at the points K, L,
and from K, L let KC, LM be drawn at right angles to BH, BG and carried
through to the points A, E.
Then, since in the circle ABC a straight line AC cuts a straight line BH into
two equal parts and at right angles, the centre of the circle ABC is on AC
[III. 1, Por.]
Again, since in the same circle ABC a straight line NO cuts a straight line BG into two equal parts and at right angles, the centre of the circle ABC is on NO.
But it was also proved to be on AC, and the straight lines AC, NO meet at no
point except at P; therefore the point P is the centre of the circle ABC.
Similarly we can prove that P is also the centre of the circle DEF; therefore
the two circles ABC, DEF which cut one another have the same centre P: which
is impossible .
Therefore, etc.
Q.E.D.