Question 193
In isosceles trapezoid $ABCD$, parallel bases $\overline{AB}$ and $\overline{CD}$ have lengths $500$ and $650$, respectively, and $AD = BC = 333$. The angle bisectors of $\angle A$ and $\angle D$ meet at $P$, and the angle bisectors of $\angle B$ and $\angle C$ meet at $Q$. Find $PQ$.