Question 135
Let $ABC$ be a triangle with incenter $I$ such that $|AB| \lt |BC| \lt |CA|$. Let $H_A, H_B, H_C$ be the orthocenters of the triangles $IBC, IAC, IAB$, respectively. Let $K_A$ and $L_A$ be points on $BC$ such that $K_A \in H_B H_C$ and $IL_A \perp H_B H_C$. Define $K_B, L_B, K_C, L_C$, similarly. Prove that $|K_A L_A| = |K_B L_B| + |K_C L_C|$.