Prove that the line joining the midpoint of parallel sides of a trapezium passes through the point of intersection of diagonals.
I want to use theorems in geometry to solve this question.
The method using vectors is given here.
Let $ABCD$ be the trapezium and let $O$ be the point of intersection of diagonals. I need to prove that one of the lines through $O$ can pass through both midpoints of adjacent sides. Let $AB$ and $CD$ be the parallel lines.
I started by considering a line through $O$ that passes through midpoint $E$ of $AB$. I need to show that if the line is extended, it also passes through midpoint $F$ of $CD$.
I tried using similarity of triangles but it did not help much.