Problem

Source: Peru Cono Sur TST 2021 #3

Tags: geometry, circumcircle



Let $ABC$ be a triangle and $D$ is a point in $BC$. The line $DA$ cuts the circumcircle of $ABC$ in the point $E$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Let $F=MN\cap AD$ and $G\neq F$ is the point of intersection of the circumcircles of $\triangle DNF$ and $\triangle ECF$. Prove that $B,F$ and $G$ are collinears.