A, B are two points inside a given circle C. Show that there are infinitely many circles through A, B which lie entirely inside C.
Solution
Let O be the center of the circle C. The perpendicular bisector of AB must meet the side OA or the side OB. wlog let it meet the side OB. If P is any point on the segment OB, then the circle center P radius PB certainly lies entirely inside C. So this is true in particular if P lies on the perpendicular bisector of AB. But in this case B also lies on the small circle. So we have found one circle.
But if we now take any point on the perpendicular bisector sufficiently close to P then the circle center P through A (and hence also B) will also lie entirely inside C.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03