Problem

Source: ELMO 2014 Shortlist G1, by Sammy Luo

Tags: geometry, circumcircle, Asymptote, geometric transformation, homothety, blogs, conics



Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear. Proposed by Sammy Luo