Let $ABC$ be a triangle. We construct squares $ABST$ and $ACUV$ with centers $O_1$ and $O_2$, respectively, as shown. Let $M$ be the midpoint of $\overline{BC}$.
(a) Prove that $\overline{BV}$ and $\overline{CT}$ are equal in length and perpendicular.
(b) Prove that $\overline{O_1 M}$ and $\overline{O_2 M}$ are equal in length and perpendicular.
I tried drawing diagonals of the squares through their centers but that didn't help much. I'm out of ideas and stuck. All solutions are highly appreciated!