Question 5
Let $ABC$ be an acute, non-isosceles triangle inscribed in a circle with center $O$ and radius $R$. A variable line $d$ is always perpendicular to $OA$ and meets rays $AB$ and $AC$ at $M$ and $N$, respectively. Suppose that lines $BN$ and $CN$ intersect at $K$, and that line $AK$ meets line $BC$.
1. Let $P=AK\cap BC$. Prove that the circumcircle of triangle $MNP$ always passes through a fixed point as $d$ varies.
2. Let $H$ be the orthocenter of triangle $AMN$. Write $BC=a$, and let $l$ be the distance from $A$ to line $HK$. Prove that $HK$ always passes through the orthocenter of triangle $ABC$. Deduce that $l\le\sqrt{4R^2-a^2}$, and determine exactly when equality holds.
1. Let $P=AK\cap BC$. Prove that the circumcircle of triangle $MNP$ always passes through a fixed point as $d$ varies.
2. Let $H$ be the orthocenter of triangle $AMN$. Write $BC=a$, and let $l$ be the distance from $A$ to line $HK$. Prove that $HK$ always passes through the orthocenter of triangle $ABC$. Deduce that $l\le\sqrt{4R^2-a^2}$, and determine exactly when equality holds.