Question 100
In acute $\triangle ABC$ denote by $M$ and $N$ the midpoints of the altitudes $BB_1$ and $CC_1$, respectively, $P = AM \cap CC_1$ and $Q = AN \cap BB_1$. Prove that:
a) the points $M, N, P$ and $Q$ are concyclic;
b) if the points $B, C, P$ and $Q$ are concyclic then $\triangle ABC$ is isosceles.
a) the points $M, N, P$ and $Q$ are concyclic;
b) if the points $B, C, P$ and $Q$ are concyclic then $\triangle ABC$ is isosceles.