Question 86
(Stoyan Boev) The incircle of an acute $\triangle ABC$ touches the sides $AB$, $BC$ and $CA$ at points $P$, $Q$ and $R$, respectively. The orthocenter $H$ of $\triangle ABC$ lies on the segment $QR$.
a) Prove that $PH \perp QR$.
b) Let $I$ and $O$ be the incenter and circumcenter of $\triangle ABC$, and $N$ the common point of $AB$ and the excircle to this side. Prove that the points $I$, $O$ and $N$ are collinear.
a) Prove that $PH \perp QR$.
b) Let $I$ and $O$ be the incenter and circumcenter of $\triangle ABC$, and $N$ the common point of $AB$ and the excircle to this side. Prove that the points $I$, $O$ and $N$ are collinear.