Question 6
Let $ABC$ be a triangle and $N$ be a point that differs from $A$, $B$ and $C$. Let $A_b$ be the reflection of $A$ through $NB$, and $B_a$ be the reflection of $B$ through $NA$. We define $B_c$, $C_b$, $A_c$ and $C_a$ similarly. Let $m_a$ be the line passing through $N$ and perpendicular to $B_cC_b$. Define $m_b$, $m_c$ similarly.
a) Assume that $N$ is the orthocenter of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through each bisector of angles $\angle BNC$, $\angle CNA$ and $\angle ANB$ are coincident.
b) Assume that $N$ is the nine-point center of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through the lines $BC$, $CA$ and $AB$ concur.
a) Assume that $N$ is the orthocenter of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through each bisector of angles $\angle BNC$, $\angle CNA$ and $\angle ANB$ are coincident.
b) Assume that $N$ is the nine-point center of triangle $ABC$, show that the respective reflections of the lines $m_a$, $m_b$ and $m_c$ through the lines $BC$, $CA$ and $AB$ concur.