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Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2020 Philippine MO
4
4
Part of
2020 Philippine MO
Problems
(1)
Right-angle revelation
Source: Philippine Mathematical Olympiad 2020/4
1/19/2020
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle with circumcircle
Γ
\Gamma
Γ
and
D
D
D
the foot of the altitude from
A
A
A
. Suppose that
A
D
=
B
C
AD=BC
A
D
=
BC
. Point
M
M
M
is the midpoint of
D
C
DC
D
C
, and the bisector of
∠
A
D
C
\angle ADC
∠
A
D
C
meets
A
C
AC
A
C
at
N
N
N
. Point
P
P
P
lies on
Γ
\Gamma
Γ
such that lines
B
P
BP
BP
and
A
C
AC
A
C
are parallel. Lines
D
N
DN
D
N
and
A
M
AM
A
M
meet at
F
F
F
, and line
P
F
PF
PF
meets
Γ
\Gamma
Γ
again at
Q
Q
Q
. Line
A
C
AC
A
C
meets the circumcircle of
△
P
N
Q
\triangle PNQ
△
PNQ
again at
E
E
E
. Prove that
∠
D
Q
E
=
9
0
∘
\angle DQE = 90^{\circ}
∠
D
QE
=
9
0
∘
.
PMO
geometry