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F is circumcenter of DMN if B is circumcenter of FMN

Source: 2022 Dutch IMO TST 2.1

December 3, 2022

geometrycircumcircle

Problem Statement

Consider an acute triangle

ABC

with

|AB| > |CA| > |BC|

. The vertices

D, E

, and

F

are the base points of the altitudes from

A, B

, and

C

, respectively. The line through F parallel to

DE

intersects

BC

in

M

. The angular bisector of

\angle MF E

intersects

DE

in

N

. Prove that

F

is the circumcentre of

\vartriangle DMN

if and only if

B

is the circumcentre of

\vartriangle FMN

.

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