Question 6
On the circumcircle of triangle$ABC$, let$M,N,P,Q$ be the midpoints of arcs$BAC,CBA,BC,AC$, respectively; the two-letter arcs are the arcs not containing the third vertex. The$A$-excircle touches$BC$ at$A_1$ and the extensions of$AC,AB$. The$B$-excircle touches$AC$ at$B_1$ and the extensions of$BA,BC$. Suppose$A_1$ lies on segment$NP$. Prove that$B_1$ lies on segment$MQ$.
