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Geometry with orthocenters

Source: Moldova TST 2020

March 7, 2020

geometryorthocenter

Problem Statement

Let

\Delta ABC

be an acute triangle and

H

its orthocenter.

B_1

and

C_1

are the feet of heights from

B

and

C

,

M

is the midpoint of

AH

. Point

K

is on the segment

B_1C_1

, but isn't on line

AH

. Line

AK

intersects the lines

MB_1

and

MC_1

in

E

and

F

, the lines

BE

and

CF

intersect at

N

. Prove that

K

is the orthocenter of

\Delta NBC

.

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