The question is
Prove that the lines joining the midpoints of opposite sides of a quadrilateral and the line joining the midpoints of the diagonals of the quadrilateral are concurrent.
I have seen other posts on this, and most of them tend to use vectors or coordinate geometry.
For example, this post: Prove that in a quadrilateral, the lines joining the midpoints of the opposite sides and the midpoints of the diagonals are concurrent
I would like to understand this problem in a way using Euclidean Geometry (Lines, Triangles, Circles, etc.). What I have tried is that the quadrilateral formed by the midpoints of opposite sides is a parallelogram. But I can't seem to get much further with this. I understood the analytical geometry part of this, but my study requires this question to be done only by Euclidean Geometry (as I'm preparing for Math Olympiad, and they tend to give extra credits to elegant solutions)
Any help would be appreciated.