CN-LEAGUE 2009 Problem 1

Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$ , intersecting $\omega$ at $P$ (different from $C$ ). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$ .

1. Prove that $MP\cdot MT=NP\cdot NT$ ;
2. Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$ . Prove that $Q,I,J,T$ are concyclic.