KöMaL problem A, 2023

Let $ABC$ be a triangle and $O$ be its circumcenter. Let $D$, $E$ and $F$ be the respective tangent points of the incircle of $\triangle ABC$, and sides $BC$, $CA$ and $AB$. Let $M$ and $N$ be the respective midpoints of sides $AB$ and $AC$. Let $M'$ and $N'$ be the respective reflections of points $M$ and $N$ across lines $DE$ and $DF$. Let lines $CM'$ and $BN'$ intersect lines $DE$ and $DF$ at points $H$ and $J$, respectively.