Winter Mathematical Competition

In $△ABC$ with $\angle ACB=60^{\circ }$ let $A{A}_1$ and $B{B}_1$ ($A_1\in BC$, $B_1\in AC$) be the bisectors of $\angle BAC$ and $\angle ABC$. The line $A_1{B}_1$ meets the circumcircle of $△ABC$ at points $A_2$ and $B_2$.

a) If $O$ and $I$ are the circumcenter and the incenter of $△ABC$ prove that $OI$ is parallel to $A_1{B}_1$.

b) If $R$ is the midpoint of the arc $\widehat{AB}$, not containing point $C$, and $P$ and $Q$ are the midpoints of $A_1{B}_1$ and $A_2{B}_2$, respectively, prove that $RP=RQ$.