IMO 1999 Bucharest geometry problem
Two circles and are contained inside the circle G, and are tangent to G at the
distinct points M and N, respectively. passes through the center of . The line
passing through the two points of intersection of and meets G at A and B.
The lines MA and MB meet at C and D, respectively.
Prove that CD is tangent to
Here it is a proof without (many) words.
Homotopy of center M which maps C in A transform in .so passes in so line passes in line and is parallel to . As is orthogonal to we have the same relation orthogonal to .
Inversion of pole B which maps P in Q , transforms in , transforms in and transforms to common tangent line so is a trapezoid with so so is tangent to