VTMO 2026 - Grade 9 - Final - SET 12 - Self-study Question 1

Let $ABC$ be a non-isosceles triangle whose shortest side is $BC$. Choose $D$ on $AB$ and $E$ on $AC$ with $BD=CE=BC$. Let $M=BE\cap CD$, let $MH\perp BC$, and set $K=AM\cap BC$.

a) Prove $BH=\dfrac{BC+BA-AC}{2}$.

b) If $AB+AC=3BC$, prove $CK=BH$.