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Iran Team Selection Test
2022 Iran Team Selection Test
4
4
Part of
2022 Iran Team Selection Test
Problems
(1)
Iran TST P4
Source: Iranian TST 2022 problem 4
4/2/2022
Cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
with circumcenter
O
O
O
is given. Point
P
P
P
is the intersection of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Let
M
M
M
and
N
N
N
be the midpoint of the sides
A
D
AD
A
D
and
B
C
BC
BC
, respectively. Suppose that
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
be the circumcircle of triangles
A
D
P
ADP
A
D
P
,
B
C
P
BCP
BCP
and
O
M
N
OMN
OMN
, respectively. The intersection point of
ω
1
\omega_1
ω
1
and
ω
3
\omega_3
ω
3
, which is not on the arc
A
P
D
APD
A
P
D
of
ω
1
\omega_1
ω
1
, is
E
E
E
and the intersection point of
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
, which is not on the arc
B
P
C
BPC
BPC
of
ω
2
\omega_2
ω
2
, is
F
F
F
. Prove that
O
F
=
O
E
OF=OE
OF
=
OE
.Proposed by Seyed Amirparsa Hosseini Nayeri
geometry