Question 6
Let $ABC$ be an acute scalene triangle. Let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. The circle with diameter $AD$ meets $DE,DF$ again at $M,N$, respectively. Let $P\in AB$ and $Q\in AC$ satisfy $NP\perp AB$, $MQ\perp AC$. Let $(I)$ be the circumcircle of $APQ$.
a) Prove that $(I)$ is tangent to $EF$.
b) Let $T$ be the tangency point, and let $DT$ meet $MN$ at $K$. Let $L$ be the reflection of $A$ across $MN$. Prove that the circumcircle of $DKL$ passes through $EF\cap MN$.
Editorial note: the circle in (b) is undefined for an allowed degenerate configuration; the solution explains this source issue.
a) Prove that $(I)$ is tangent to $EF$.
b) Let $T$ be the tangency point, and let $DT$ meet $MN$ at $K$. Let $L$ be the reflection of $A$ across $MN$. Prove that the circumcircle of $DKL$ passes through $EF\cap MN$.
Editorial note: the circle in (b) is undefined for an allowed degenerate configuration; the solution explains this source issue.