MathNet 2018-part-01 — 181 problems (EN–VI) Question 111

The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y = x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals $AC$ and $BD$ respectively.
Find the length of the projection of the segment $MN$ on the axis $Oy$.