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Turkey MO (2nd round)
2020 Turkey MO (2nd round)
2
2
Part of
2020 Turkey MO (2nd round)
Problems
(1)
Four points on AP
Source: Turkey National Mathematical Olympiad 2020 P2
3/8/2021
Let
P
P
P
be an interior point of acute triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
, which is different from the orthocenter. Let
D
D
D
and
E
E
E
be the feet of altitudes from
A
A
A
to
B
P
BP
BP
and
C
P
CP
CP
, and let
F
F
F
and
G
G
G
be the feet of the altitudes from
P
P
P
to sides
A
B
AB
A
B
and
A
C
AC
A
C
. Denote by
X
X
X
the midpoint of
[
A
P
]
[AP]
[
A
P
]
, and let the second intersection of the circumcircles of triangles
Δ
D
F
X
\Delta DFX
Δ
D
FX
and
Δ
E
G
X
\Delta EGX
Δ
EGX
lie on
B
C
BC
BC
. Prove that
A
P
AP
A
P
is perpendicular to
B
C
BC
BC
or
∠
P
B
A
=
∠
P
C
A
\angle PBA = \angle PCA
∠
PB
A
=
∠
PC
A
.
geometry