Chapter 12.4B, Problem 22A

$ABCD$ is a convex quadrilateral and $M,N,P,Q$ are the midpoints of its sides as shown. The diagonals $\overline{BD}$ and $\overline{AC}$ intersect at $T$, $\overline{NP}$ intersect $\overline{AC}$ at $S$, and $\overline{BD}$ intersect $\overline{MN}$ at $V$.

a. Prove that $NSTV$ is a parallelogram.

b. Complete the following statements and prove them.

i. $MNPQ$ is a rectangle if, and only if, $\overline{AC}$ and $\overline{BD}$ are $\text{\_}$.

ii. $MNPQ$ is a square if, and only if, $\overline{AC}$ and $\overline{BD}$ are $\text{\_}$.