Let Q be the intersection of any pair of circles orthogonal to both C and M. Invert with respect to Q. Then these two circles invert to a pair of intersecting lines. Call the point of intersection R. The inverse of C is a circle perpendicular to both of the lines, and so with centre on both lines. That is, it is a circle centred at R. Similarly the inverse of m is a circle centred at R. Then any circle perpendicular to both m and C must invert to a circle or line perpendicular to each of the two concentric circles, that is, to a line through their common centre at R. So any circle orthogonal to both C and m must pass through both Q and R', the inverse of R.

This proof ahows that there are 2 such fixed points. Note that I have nowhere used the fact that m is a straight line, so that the proof holds for a general pair of nonintersecting circles.

Thankyou

sfwc
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