Line $\ell$ is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the remaining two.
Problem
Source: Oral Moscow Geometry Olympiad 2015 grades 10-11 p2
Tags: geometry, perpendicular bisector, midpoint