ELMO SL 2014 G1

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