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Indonesia MO
2018 Indonesia MO
8
8
Part of
2018 Indonesia MO
Problems
(1)
Excircle tangency points lie on the circumcircle
Source: Indonesian National Science Olympiad 2018, Mathematics P8
7/6/2018
Let
I
,
O
I, O
I
,
O
be the incenter and circumcenter of the triangle
A
B
C
ABC
A
BC
respectively. Let the excircle
ω
A
\omega_A
ω
A
of
A
B
C
ABC
A
BC
be tangent to the side
B
C
BC
BC
on
N
N
N
, and tangent to the extensions of the sides
A
B
,
A
C
AB, AC
A
B
,
A
C
on
K
,
M
K, M
K
,
M
respectively. If the midpoint of
K
M
KM
K
M
lies on the circumcircle of
A
B
C
ABC
A
BC
, prove that
O
,
I
,
N
O, I, N
O
,
I
,
N
are collinear.
geometry