Question 111
Let $ABC$ be an acute, non-isosceles triangle with altitudes $AD$, $BE$ and $CF$. The circle ($AD$) intersects $DE$, $DF$ at $M$, $N$. Let $P$, $Q$ be the points on $AB$, $AC$ respectively such that $NP$ is perpendicular to $AB$ and $MQ$ is parallel to $AC$. Let ($I$) be the circumcircle of triangle $APQ$.
a) Prove that ($I$) is tangent to $EF$.
b) Let $T$ be the tangency point of the circumcircle of triangle $APQ$ with $EF$, $K$ be the intersection of $DT$, $MN$ and $L$ be the reflection of $A$ through $MN$. Prove that the circumcircle of triangle $DKL$ passes through the intersection of $MN$, $EF$.
a) Prove that ($I$) is tangent to $EF$.
b) Let $T$ be the tangency point of the circumcircle of triangle $APQ$ with $EF$, $K$ be the intersection of $DT$, $MN$ and $L$ be the reflection of $A$ through $MN$. Prove that the circumcircle of triangle $DKL$ passes through the intersection of $MN$, $EF$.