# IMO 1999 Bucharest geometry problem

Submitted by Structure on Thu, 01/08/2009 - 21:15.

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