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
be joined and bisected at the points K
, and from K
be drawn at right angles to BH
and carried through to the points A
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
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. [III. 5]
Therefore etc. Q. E. D.