Question 36
Let $O$ be the center of the equilateral triangle $ABC$. Pick two points $P_1$ and $P_2$ other than $B, O, C$ on the circle $\odot(BOC)$ so that on this circle $B, P_1, P_2, O, C$ are placed in this order. Extensions of $BP_1$ and $CP_1$ intersect respectively with side $CA$ and $AB$ at points $R$ and $S$. Line $AP_1$ and $RS$ meets at point $Q_1$. Analogously point $Q_2$ is defined. Let $\odot(OP_1Q_1)$ and $\odot(OP_2Q_2)$ meet again at point $U$ other than $O$.
Prove that $2\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ$.
Prove that $2\angle Q_2UQ_1 + \angle Q_2OQ_1 = 360^\circ$.