Question 23
Let $ABC$ be a triangle with incentre $I$ such that $AB \lt AC \lt BC$. The second intersections of $AI$, $BI$, and $CI$ with the circumcircle of triangle $ABC$ are $M_A$, $M_B$, and $M_C$, respectively. Lines $AI$ and $BC$ intersect at $D$ and lines $BM_C$ and $CM_B$ intersect at $X$. Suppose the circumcircles of triangles $XM_B M_C$ and $XBC$ intersect again at $S \neq X$. Lines $BX$ and $CX$ intersect the circumcircle of triangle $SXM_A$ again at $P \neq X$ and $Q \neq X$, respectively.
Prove that the circumcentre of triangle SID lies on PQ.
Prove that the circumcentre of triangle SID lies on PQ.