Euclid's Elements Book III Proposition 10

A circle does not cut a circle at more than two points.

For, if

possible, let the

circle ABC

cut the circle DEF at more points than two,

namely B, G, F, and H.

Join BH and BG, and bisect them at the points K and L. (I.10)

Draw KC and LM from K and L at right angles to BH and BG, (I.11)

and carry them through to the points A and 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 center of the circle ABC lies on AC.

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 center of the circle ABC lies on NO. (III.1)

But it was also proved to lie on AC, and the straight lines AC and NO meet at no point except at P,

therefore the point P is the center of the circle ABC.

Similarly we can prove that P is also the center of the circle DEF,

therefore the two circles ABC and DEF which cut one another have the same center P, which is impossible. (III.5)

Therefore a circle does not cut a circle at more than two points.

Q.E.D.

Proposition 9 <-- Proposition 10 --> Proposition 11