MathNet 2026 — Part 07 Question 195

Let $ABC$ be a triangle with $AB \lt AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that $K$ is the midpoint of $CL$.