Question 115
As shown in the Fig. 1, $M$, $N$ are the midpoints of arcs $\widehat{BC}$, $\widehat{AC}$ respectively, which are on the circumscribed circle $\Gamma$ of an acute triangle $\triangle ABC$ ($\angle A \lt \angle B$). Through point $C$ draw $PC \parallel MN$, intercepting the circle $\Gamma$ at point $P$. $I$ is the inner center of $\triangle ABC$. Extend line $PI$ to intercept $\Gamma$ at point $T$.
Fig. 1
(1) Prove that $MP \times MT = NP \times NT$;
(2) For an arbitrary point $Q (\neq A, T, B)$ on arc $\widehat{AB}$ (not containing $C$), denote the inner centers of $\triangle AQC$, $\triangle QCB$ by $I_1, I_2$, respectively. Prove that $Q, I_1, I_2, T$ are concyclic.
Fig. 1
(1) Prove that $MP \times MT = NP \times NT$;
(2) For an arbitrary point $Q (\neq A, T, B)$ on arc $\widehat{AB}$ (not containing $C$), denote the inner centers of $\triangle AQC$, $\triangle QCB$ by $I_1, I_2$, respectively. Prove that $Q, I_1, I_2, T$ are concyclic.