IMO 1979 LL POL51

Origin: POL

Problem

Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be circles satisfying the following conditions: S1 is tangent to CA and AB, S2 is tangent to S1, AB, and BC, S3 is tangent to S2, BC, and CA, \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot S7 is tangent to S6, CA and AB. Prove that the circles S1 and S7 coincide.