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J, O, M colinear iff M is midpoint

Source: JOM 2023 P3

February 20, 2023

geometry

Problem Statement

Given an acute triangle

ABC

with

AB<AC

, let

D

be the foot of altitude from

A

to

BC

and let

M\neq D

be a point on segment

BC

.

\,J

and

K

lie on

AC

and

AB

respectively such that

K,J,M

lies on a common line perpendicular to

BC

. Let the circumcircles of

\triangle ABJ

and

\triangle ACK

intersect at

O

. Prove that

J,O,M

are collinear if and only if

M

is the midpoint of

BC

.
Proposed by Wong Jer Ren

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