RU-ORAL-GEOMETRY 2017

On side $AB$ of triangle $ABC$ is marked point $K$ such that $AB = CK$ . Points $N$ and $M$ are the midpoints of $AK$ and $BC$ , respectively. The segments $NM$ and $CK$ intersect in point $P$ . Prove that $KN = KP$ .

An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. Circles with centers $O_1$ and $O_2$ are inscribed in triangles $ABC$ and $ABD$ . Prove that line $O_1O_2$ is perpendicular on $BC$ .

Points $M$ and $N$ are the midpoints of sides $AB$ and $CD$ , respectively of quadrilateral $ABCD$ . It is known that $BC // AD$ and $AN = CM$ . Is it true that $ABCD$ is parallelogram?