MathNet 2019-part-05 — 184 problems (EN–VI) Question 27

Let $ABC$ be a non-isosceles triangle, and let $I$ be its incenter. Let $\gamma_A$ be the circle through $I$, tangent to $AB$ and $AC$ and crossing the segment $AI$, let $\gamma_B$ be the circle through $I$ tangent to $BC$ and $BA$ and crossing the segment $BI$, and let $\gamma_C$ be the circle through $I$, tangent to $CA$ and $CB$ and crossing the segment $CI$. The circles $\gamma_B$ and $\gamma_C$ cross again at $A'$, the circles $\gamma_C$ and $\gamma_A$ cross again at $B'$, and the circles $\gamma_A$ and $\gamma_B$ cross again at $C'$. Prove that the circles $AA'I, BB'I$ and $CC'I$ cross again at a point different from $I$.
IMO 1997, Longlist