ZA-SAMO 2021 Problem 4

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$ . Let $K$ be the circle through $A$ with centre $B$ . Let $D$ be the other intersection of $K$ and $AC$ . Let $K$ intersect the circumcircle of $BCD$ again at $E$ . If $F$ is the intersection of $DE$ and $BH$ , show that $BD$ is tangent to the circle through $D$ , $F$ , and $H$ .