4. Let $A B C D$ be a convex quadrilateral with $A B=5, B C=6, C D=7$, and $D A=8$. Let $M, P, N, Q$ be the midpoints of sides $A B, B C, C D, D A$ respectively. Compute $M N^{2}-P Q^{2}$.
Answer: $\square$
Draw in the diagonals of the quad and use the median formula three times to get $M N^{2}$ in terms of the diagonals. Do the same for $P Q^{2}$ and subtract, the diagonal length terms disappear and the answer is
$$
\frac{B C^{2}+D A^{2}-A B^{2}-C D^{2}}{2}=13
$$