MathDB

5

Part of 2021 Vietnam TST

Problems(1)

Goes through fixed points

Source: Vietnam TST 2021 P5

4/2/2021
Given a fixed circle (O)(O) and two fixed points B,CB, C on that circle, let AA be a moving point on (O)(O) such that △ABC\triangle ABC is acute and scalene. Let II be the midpoint of BCBC and let AD,BE,CFAD, BE, CF be the three heights of △ABC\triangle ABC. In two rays FA→,EA→\overrightarrow{FA}, \overrightarrow{EA}, we pick respectively M,NM,N such that FM=CE,EN=BFFM = CE, EN = BF. Let LL be the intersection of MNMN and EFEF, and let G≠LG \neq L be the second intersection of (LEN)(LEN) and (LFM)(LFM).
a) Show that the circle (MNG)(MNG) always goes through a fixed point.
b) Let ADAD intersects (O)(O) at K≠AK \neq A. In the tangent line through DD of (DKI)(DKI), we pick P,QP,Q such that GP∥AB,GQ∥ACGP \parallel AB, GQ \parallel AC. Let TT be the center of (GPQ)(GPQ). Show that GTGT always goes through a fixed point.
geometry