BMO Round 1, 2023

In an acute, non-isosceles triangle $ABC$ the midpoints of $AC$ and $AB$ are $B_1$ and $C_1$ respectively. A point $D$ lies on $BC$ with $C$ between $B$ and $D$. The point $F$ is such that $\angle AFC$ is a right angle and $\angle DCF = \angle FCA$. The point $G$ is such that $\angle AGB$ is a right angle and $\angle CBG = \angle GBA$. Prove that $B_1$, $C_1$, $F$ and $G$ are collinear.