MathDB

3

Part of 2021 Vietnam TST

Problems(1)

Reflection and concurrence

Source: Vietnam TST 2021 P3

4/1/2021
Let ABCABC be a triangle and NN be a point that differs from A,B,CA,B,C. Let AbA_b be the reflection of AA through NBNB, and BaB_a be the reflection of BB through NANA. Similarly, we define Bc,Cb,Ac,CaB_c, C_b, A_c, C_a. Let mam_a be the line through NN and perpendicular to BcCbB_cC_b. Define similarly mb,mcm_b, m_c.
a) Assume that NN is the orthocenter of △ABC\triangle ABC, show that the respective reflection of ma,mb,mcm_a, m_b, m_c through the bisector of angles ∠BNC,∠CNA,∠ANB\angle BNC, \angle CNA, \angle ANB are the same line. b) Assume that NN is the nine-point center of △ABC\triangle ABC, show that the respective reflection of ma,mb,mcm_a, m_b, m_c through BC,CA,ABBC, CA, AB concur.
geometrygeometric transformationreflection