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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2018 Korea Junior Math Olympiad
3
3
Part of
2018 Korea Junior Math Olympiad
Problems
(1)
incenters of half the triangle
Source: KJMO 2018 p3
7/26/2019
Let there be a scalene triangle
A
B
C
ABC
A
BC
, and denote
M
M
M
by the midpoint of
B
C
BC
BC
. The perpendicular bisector of
B
C
BC
BC
meets the circumcircle of
A
B
C
ABC
A
BC
at point
P
P
P
, on the same side with
A
A
A
with respect to
B
C
BC
BC
. Let the incenters of
A
B
M
ABM
A
BM
and
A
M
C
AMC
A
MC
be
I
,
J
I,J
I
,
J
, respectively. Let
∠
B
A
C
=
α
\angle BAC=\alpha
∠
B
A
C
=
α
,
∠
A
B
C
=
β
\angle ABC=\beta
∠
A
BC
=
β
,
∠
B
C
A
=
γ
\angle BCA=\gamma
∠
BC
A
=
γ
. Find
∠
I
P
J
\angle IPJ
∠
I
P
J
.
geometry
incenter