Question 30
In an isosceles triangle $ABC$ with $|AB| = |AC|$, points $M$ and $N$ are the midpoints of the sides $\overline{AB}$ and $\overline{BC}$, respectively. The circle circumscribed to the triangle $AMC$ meets the line $AN$ at point $P$ different from $A$. The line passing through $P$ parallel to the side $BC$ meets the circle circumscribed to the triangle $ABC$ at points $B_1$ and $C_1$. Prove that the triangle $AB_1C_1$ is equilateral.