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Contests
International Contests
EGMO
2023 EGMO
6
6
Part of
2023 EGMO
Problems
(1)
A Familiar Point
Source: EGMO 2023/6
4/16/2023
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
Ω
\Omega
Ω
. Let
S
b
S_b
S
b
and
S
c
S_c
S
c
respectively denote the midpoints of the arcs
A
C
AC
A
C
and
A
B
AB
A
B
that do not contain the third vertex. Let
N
a
N_a
N
a
denote the midpoint of arc
B
A
C
BAC
B
A
C
(the arc
B
C
BC
BC
including
A
A
A
). Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
. Let
ω
b
\omega_b
ω
b
be the circle that is tangent to
A
B
AB
A
B
and internally tangent to
Ω
\Omega
Ω
at
S
b
S_b
S
b
, and let
ω
c
\omega_c
ω
c
be the circle that is tangent to
A
C
AC
A
C
and internally tangent to
Ω
\Omega
Ω
at
S
c
S_c
S
c
. Show that the line
I
N
a
IN_a
I
N
a
, and the lines through the intersections of
ω
b
\omega_b
ω
b
and
ω
c
\omega_c
ω
c
, meet on
Ω
\Omega
Ω
.
EGMO
geometry
EGMO 2023