Problem

Source: Romania JBMO TST 2024 Day 4 P2

Tags: geometry, ROMANIA JBMO TST



Let $ABC$ be a scalene triangle, with circumcircle $\omega$ and incentre $I.{}$ The tangent line at $C$ to $\omega$ intersects the line $AB$ at $D.{}$ The angle bisector of $BDC$ meets $BI$ at $P{}$ and $AI{}$ at $Q{}.$ Let $M{}$ be the midpoint of the segment $PQ.$ Prove that the line $IM$ passes through the middle of the arc $ACB$ of $\omega.$ Dana Heuberger