Question 130
The bisectors of angles $\angle A$ and $\angle C$ of a convex quadrilateral $ABCD$ meet at the point $E$, and the bisectors of angles $\angle B$ and $\angle D$ meet at the point $F$ ($E$ and $F$ lie in the interior of $ABCD$). The point $M$ is the midpoint of the segment $EF$. The points $H_1, H_2, H_3$ and $H_4$ are the feet of the perpendiculars from $M$ to the sides $AB, BC, CD$ and $AD$ respectively.
Prove that $MH_1 + MH_3 = MH_2 + MH_4$.
Prove that $MH_1 + MH_3 = MH_2 + MH_4$.