MathNet 2015-part-06 — 191 problems (EN–VI) Question 171

Let $A_1$ and $C_1$ be the points on the sides $BC$ and $AB$ of the triangle $ABC$ respectively, so that $AA_1 = CC_1$. Segments $AA_1$ and $CC_1$ meet at the point $F$. Prove that if $\angle CFA_1 = 2\angle ABC$ then $AA_1 = AC$.