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National and Regional Contests
Azerbaijan Contests
JBMO TST - Azerbaijan
2018 Azerbaijan JBMO TST
1
1
Part of
2018 Azerbaijan JBMO TST
Problems
(1)
Geometry with orthocenter and perpendiculars
Source: Azerbaijan JBMO TST 2018, D2 P1
8/1/2023
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle. Let us denote the foot of the altitudes from the vertices
A
,
B
A, B
A
,
B
and
C
C
C
to the opposite sides by
D
,
E
D, E
D
,
E
and
F
,
F,
F
,
respectively, and the intersection point of the altitudes of the triangle
A
B
C
ABC
A
BC
by
H
.
H.
H
.
Let
P
P
P
be the intersection of the line
B
E
BE
BE
and the segment
D
F
.
DF.
D
F
.
A straight line passing through
P
P
P
and perpendicular to
B
C
BC
BC
intersects
A
B
AB
A
B
at
Q
.
Q.
Q
.
Let
N
N
N
be the intersection of the segment
E
Q
EQ
EQ
with the perpendicular drawn from
A
.
A.
A
.
Prove that
N
N
N
is the midpoint of segment
A
H
.
AH.
A
H
.
geometry