ELMO SL 2019 G5

Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$ , let $B_1, C_1$ be variable points on $AB,AC$ , respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$ . Denote the circumcircle of $AB_1C_1$ by $\omega$ . Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$ , then $\omega$ passes through a fixed point other than $A$ .

Proposed by Max Jiang