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National and Regional Contests
Poland Contests
Poland - Second Round
2017 Poland - Second Round
2
2
Part of
2017 Poland - Second Round
Problems
(1)
Orthogonal projections and areas
Source: Polish MO
2/25/2017
In an acute triangle
A
B
C
ABC
A
BC
the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
crosses
B
C
BC
BC
at
D
D
D
. Points
P
P
P
and
Q
Q
Q
are orthogonal projections of
D
D
D
on lines
A
B
AB
A
B
and
A
C
AC
A
C
. Prove that
[
A
P
Q
]
=
[
B
C
Q
P
]
[APQ]=[BCQP]
[
A
PQ
]
=
[
BCQP
]
if and only if the circumcenter of
A
B
C
ABC
A
BC
lies on
P
Q
PQ
PQ
.
geometry