PAMO 2022 Problem 1

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ , and $AB$ its shortest side. Let $H$ be the orthocenter of $ABC$ . Let $\Gamma$ be the circle with center $B$ and radius $BA$ . Let $D$ be the second point where the line $CA$ meets $\Gamma$ . Let $E$ be the second point where $\Gamma$ meets the circumcircle of the triangle $BCD$ . Let $F$ be the intersection point of the lines $DE$ and $BH$ .

Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$ .