SHARYGIN 2006 Problem 93

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$ , a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$ . We similarly construct $B'$ and $C'$ . Prove that $A', B',C'$ lie on one straight line.