Question 50
Let $\triangle ABC$ be an acute triangle. Denote by $E$ and $F$ the feet of the altitudes from $B$ and $C$, respectively. Let $H$ be the intersection of $BE$ and $CF$. Let $D$ be on the same side of line $BC$ as $A$ and satisfy:
$$
\angle DBC = \angle DCB = \angle BAC.
$$
Let $N$ be the midpoint of $EF$. Prove that points $H$, $D$ and $N$ are collinear.
Figure 23
Figure-label clarification: The original Figure 23 labels the midpoint of $EF$ as $M$; this is the point called $N$ in the problem. The original image is preserved.
$$
\angle DBC = \angle DCB = \angle BAC.
$$
Let $N$ be the midpoint of $EF$. Prove that points $H$, $D$ and $N$ are collinear.
Figure 23
Figure-label clarification: The original Figure 23 labels the midpoint of $EF$ as $M$; this is the point called $N$ in the problem. The original image is preserved.