RU-ORAL-GEOMETRY 2018

Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the diagonals of the second.

Bisectors of angle $C$ and externalangle $A$ of trapezoid $ABCD$ with bases $BC$ and $AD$ intersect at point $M$ , and the bisector of angle $B$ and external angle $D$ intersect at point $N$ . Prove that the midpoint of the segment $MN$ is equidistant from the lines $AB$ and $CD$ .

On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$ , respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$ , side $BC$ and the extension of side $AC$ beyond point $C$ . Point $M$ is the midpoint of segment $EF$ . Prove that the line $MN$ is parallel to the bisector of angle $A$ .