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National and Regional Contests
Israel Contests
Israel Olympic Revenge
2024 Israel Olympic Revenge
P4
P4
Part of
2024 Israel Olympic Revenge
Problems
(1)
Line through O given concurrency
Source: 2024 Israel Olympic Revenge P4
6/20/2024
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
D
D
D
be a point inside side
B
C
BC
BC
. Let
E
E
E
be the foot from
D
D
D
to
A
C
AC
A
C
, and let
F
F
F
be a point on
A
B
AB
A
B
so that
F
E
⊥
A
B
FE\perp AB
FE
⊥
A
B
. It is given that the lines
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
concur.
M
A
,
M
B
,
M
C
M_A, M_B, M_C
M
A
,
M
B
,
M
C
are the midpoints of sides
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
respectively, and
O
O
O
is the circumcenter of
A
B
C
ABC
A
BC
. Moreover, we define
P
=
E
F
∩
M
A
M
B
,
S
=
D
E
∩
M
A
M
C
P=EF\cap M_AM_B, S=DE\cap M_AM_C
P
=
EF
∩
M
A
M
B
,
S
=
D
E
∩
M
A
M
C
. Prove that
O
,
P
,
S
O, P, S
O
,
P
,
S
are collinear.
geometry
olympic revenge