VN-TST 2021 Problem 3

Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$ . Let $A_b$ be the reflection of $A$ through $NB$ , and $B_a$ be the reflection of $B$ through $NA$ . Similarly, we define $B_c, C_b, A_c, C_a$ . Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$ . Define similarly $m_b, m_c$ .

a) Assume that $N$ is the orthocenter of $\triangle ABC$ , show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line. b) Assume that $N$ is the nine-point center of $\triangle ABC$ , show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.