MathNet 2022-part-03 — 197 problems (EN–VI) Question 43

Let $ABC$ be a triangle, let $r$ be the internal bisector of the acute angle $\widehat{BAC}$, and let $K$ be the projection of $B$ onto $r$, $L$ the projection of $K$ onto $AB$, and $D$ the reflection of $B$ in $L$. Finally, let $H$ be the foot of the altitude of triangle $ABC$ from $B$. Prove that:

a. $BH = 2 LK$;

b. $KA$ bisects the angle $\widehat{HKD}$;

c. triangle $ADH$ is isosceles.