Question 104
In $\triangle ABC$ with $\angle ACB = 60^\circ$ let $AA_1$ and $BB_1$ ($A_1 \in BC$, $B_1 \in AC$) be the bisectors of $\angle BAC$ and $\angle ABC$. The line $A_1B_1$ meets the circumcircle of $\triangle ABC$ at points $A_2$ and $B_2$.
a) If $O$ and $I$ are the circumcenter and the incenter of $\triangle ABC$ prove that $OI$ is parallel to $A_1B_1$.
b) If $R$ is the midpoint of the arc $\widehat{AB}$, not containing point $C$, and $P$ and $Q$ are the midpoints of $A_1B_1$ and $A_2B_2$, respectively, prove that $RP = RQ$.
a) If $O$ and $I$ are the circumcenter and the incenter of $\triangle ABC$ prove that $OI$ is parallel to $A_1B_1$.
b) If $R$ is the midpoint of the arc $\widehat{AB}$, not containing point $C$, and $P$ and $Q$ are the midpoints of $A_1B_1$ and $A_2B_2$, respectively, prove that $RP = RQ$.