SHARYGIN 2025 Problem 104

Let $M, H, L$ be the centroid, the orthocenter, and the Lemoine point respectively of a triangle $ABC$ . A point $S$ is such that the circles $SLH, SML$ touch $MH$ , and $L'$ is the reflection of $L$ about the circumcircle of the triangle. Prove that $SL' \parallel MH$ .

Proposed by S. Kuznetsov