IMO 1969 LL NET47

Origin: NET

Problem

Let A and B be points on the circle \gamma. A point C, different from A and B, is on the circle \gamma. Let D be the projection of the point C onto the line AB. Consider three other circles \gamma1, \gamma2, and \gamma3 with the common tangent AB: \gamma1 inscribed in the triangle ABC, and \gamma2 and \gamma3 tangent to both (the segment) CD and \gamma. Prove that \gamma1, \gamma2, and \gamma3 have two common tangents.