Question 3
Let $ABC$ be a triangle and let $N$ be a point different from $A,B,C$. Let $A_b$ be the reflection of $A$ across $NB$, and let $B_a$ be the reflection of $B$ across $NA$. Define the other two pairs $B_c,C_b$ and $C_a,A_c$ similarly. Let $m_a$ pass through $N$ and be perpendicular to $B_cC_b$, and define $m_b,m_c$ cyclically.
(a) Suppose $N$ is the orthocenter of $ABC$. Prove that the reflections of $m_a,m_b,m_c$ across the bisectors of $\angle BNC,\angle CNA,\angle ANB$, respectively, coincide.
(b) Suppose $N$ is the center of the nine-point (Euler) circle of $ABC$. Prove that the reflections of $m_a,m_b,m_c$ across $BC,CA,AB$, respectively, are concurrent at one point.
(a) Suppose $N$ is the orthocenter of $ABC$. Prove that the reflections of $m_a,m_b,m_c$ across the bisectors of $\angle BNC,\angle CNA,\angle ANB$, respectively, coincide.
(b) Suppose $N$ is the center of the nine-point (Euler) circle of $ABC$. Prove that the reflections of $m_a,m_b,m_c$ across $BC,CA,AB$, respectively, are concurrent at one point.