VMO 2023 Problem 4

Given is a triangle $ABC$ and let $D$ be the midpoint the major arc $BAC$ of its circumcircle. Let $M , N$ be the midpoints of $AB , AC$ and $J , E , F$ are the touchpoints of the incircle $(I)$ of $\triangle ABC$ with $BC, CA, AB$ . The line $MN$ intersects $JE , JF$ at $K , H$ respectively; $IJ$ intersects the circle $(BIC)$ at $G$ and $DG$ intersects $(BIC)$ at $T$ .

a) Prove that $JA$ passes through the midpoint of $HK$ and is perpendicular to $IT$ .

b) Let $R, S$ respectively be the perpendicular projection of $D$ on $AB, AC$ . Take the points $P, Q$ on $IF , IE$ respectively such that $KP$ and $HQ$ are both perpendicular to $MN$ . Prove that the three lines $MP , NQ$ and $RS$ are concurrent .