Question 170
In the plane fix two points $A, B$ ($A \neq B$). Consider a point $C$ moving in the plane such that $\widehat{ACB} = \alpha$, where $\alpha$ is a given angle ($0^\circ \lt \alpha \lt 180^\circ$). The circle with center in $I$, inscribed the triangle $ABC$ touches the sides $AB$, $BC$ and $CA$ in the points $D$, $E$ and $F$, respectively. The line $AI$ and $BI$ intersect $EF$ in $M$ and $N$, respectively. Show that:
1/ The segment $MN$ has constant length;
2/ The circumcircle of the triangle $DMN$ goes through a fix-point.
1/ The segment $MN$ has constant length;
2/ The circumcircle of the triangle $DMN$ goes through a fix-point.