MathDB

Configuration with midpoints and altitudes

Source: KoMaL A857

October 11, 2023

geometrykomal

Problem Statement

Let

ABC

be a given acute triangle, in which

BC

is the longest side. Let

H

be the orthocenter of the triangle, and let

D

and

E

be the feet of the altitudes from

B

and

C

, respectively. Let

O_1

and

G

be the midpoints of sides

AB

and

AC

, respectively.

X

is the point of intersection of lines

DF

and

EG

. Let

O_1

and

O_2

be the circumcenters of triangles

EFX

and

DGX

, respectively. Finally,

M

is the midpoint of line segment

O_1O_2

. Prove that points

X, H

and

M

are collinear.

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