Proposition III.10 — A circle does not cut a circle at more points than two.

Proof

Suppose two circles meet at three points

A

B

C

. By III.9, the centre of each circle is the unique point equidistant from any three points on its circumference — so both circles have the same centre. Then by III.5 they coincide, contradicting their being two distinct circles.

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Depends on (2)

III.5Proposition III.5If two circles cut one another, they will not have the same centre.
III.9Proposition III.9If a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the…

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