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.
Source: Sharygin 2006 finals 9.3
Tags: equal ratio, similar triangles, collinear, geometry
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.