Quadrilateral problem with midpoints of diagonals
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For an arbitrary convex quadrialteral $ABCD$:
$$AF=FC, BE=ED, GI=IH$$
Prove that points $F$, $E$ and $I$ are collinear.
I was able to solve the problem by using vectors and I think that it can be also solved analytically but I wanted some more elegant proof, more in the spirit of Euclid.