MathDB

3

Part of JOM 2023

Problems(1)

J, O, M colinear iff M is midpoint

Source: JOM 2023 P3

2/20/2023
Given an acute triangle ABCABC with AB<ACAB<AC, let DD be the foot of altitude from AA to BCBC and let M≠DM\neq D be a point on segment BCBC. J\,J and KK lie on ACAC and ABAB respectively such that K,J,MK,J,M lies on a common line perpendicular to BCBC. Let the circumcircles of △ABJ\triangle ABJ and △ACK\triangle ACK intersect at OO. Prove that J,O,MJ,O,M are collinear if and only if MM is the midpoint of BCBC.
Proposed by Wong Jer Ren
geometry