Problem

Source: ELMO Shortlist 2010, G1

Tags: geometry, circumcircle, geometric transformation, incenter, homothety, Asymptote, angle bisector



Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent. Carl Lian.