VTMO 2017 — Grade 9 — final Question 5

$ABC$ is scalene with $BC$ its shortest side. $D \in AB , E \in AC$ satisfy $BD = CE = BC$. Let $M = BE \cap CD , MH \perp BC , K = AM \cap BC$. Prove $BH = \frac{BC + BA - AC}{2}$, and if $AB + AC = 3 BC$, prove $CK = BH$.