Question 2
Let $ABCD$ be a quadrilateral with $\angle ABC \neq 90^\circ$. Let $M$ and $N$ be the midpoints of $AD$ and $CD$, respectively. Prove that the lines perpendicular to $BC$ passing through $M$ and perpendicular to $AB$ passing through $N$ and $BD$ are concurrent if and only if the diagonals $BD$ and $AC$ are perpendicular.