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Moldova Contests
Moldova Team Selection Test
2020 Moldova Team Selection Test
4
4
Part of
2020 Moldova Team Selection Test
Problems
(1)
Geometry with orthocenters
Source: Moldova TST 2020
3/7/2020
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
be an acute triangle and
H
H
H
its orthocenter.
B
1
B_1
B
1
and
C
1
C_1
C
1
are the feet of heights from
B
B
B
and
C
C
C
,
M
M
M
is the midpoint of
A
H
AH
A
H
. Point
K
K
K
is on the segment
B
1
C
1
B_1C_1
B
1
C
1
, but isn't on line
A
H
AH
A
H
. Line
A
K
AK
A
K
intersects the lines
M
B
1
MB_1
M
B
1
and
M
C
1
MC_1
M
C
1
in
E
E
E
and
F
F
F
, the lines
B
E
BE
BE
and
C
F
CF
CF
intersect at
N
N
N
. Prove that
K
K
K
is the orthocenter of
Δ
N
B
C
\Delta NBC
Δ
NBC
.
geometry
orthocenter