NMC 2023 Problem 4

Let $ABC$ be a triangle, and $M$ the midpoint of the side $BC$ . Let $E$ and $F$ be points on the sides $AC$ and $AB$ , respectively, so that $ME=MF$ . Let $D$ be the second intersection of the circumcircle of $MEF$ and the side $BC$ . Consider the lines $\ell_D$ , $\ell_E$ and $\ell_F$ through $D, E$ and $F$ , respectively, such that $\ell_D \perp BC$ , $\ell_E \perp AC$ and $\ell_F \perp AB$ . Show that $\ell_D, \ell_E$ and $\ell_F$ are concurrent.