Question 3
In the plane, let $A,B$ be fixed distinct points. A point $C$ varies so that $\angle ACB=\alpha$, where $0^\circ\lt \alpha\lt 180^\circ$ is fixed. The incircle of triangle $ABC$, with center $I$, touches $AB,BC,CA$ at $D,E,F$, respectively. Lines $AI,BI$ meet line $EF$ at $M,N$, respectively. Prove that:
1. The length $MN$ is constant.
2. The circumcircle of triangle $DMN$ always passes through a fixed point.
1. The length $MN$ is constant.
2. The circumcircle of triangle $DMN$ always passes through a fixed point.