BALKAN 2026 Problem 3

Let $ABCD$ be a parallelogram with $\angle DAB < 90^\circ$ and $AB < AD$. Let $H$ be the orthocentre of $\triangle BCD$ and $H'$ be the reflection of $H$ over line $BD$. Line $AH$ intersects the lines $BD, CD$ and $BC$ at $E, F$ and $G$ respectively. Prove that the circumcircles of $\triangle HEH'$ and $\triangle CFG$ are tangent.