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International Contests
Middle European Mathematical Olympiad
2015 Middle European Mathematical Olympiad
6
6
Part of
2015 Middle European Mathematical Olympiad
Problems
(1)
Two isogonals in incenter diagram
Source: MEMO 2015, problem T-6
8/28/2015
Let
I
I
I
be the incentre of triangle
A
B
C
ABC
A
BC
with
A
B
>
A
C
AB>AC
A
B
>
A
C
and let the line
A
I
AI
A
I
intersect the side
B
C
BC
BC
at
D
D
D
. Suppose that point
P
P
P
lies on the segment
B
C
BC
BC
and satisfies
P
I
=
P
D
PI=PD
P
I
=
P
D
. Further, let
J
J
J
be the point obtained by reflecting
I
I
I
over the perpendicular bisector of
B
C
BC
BC
, and let
Q
Q
Q
be the other intersection of the circumcircles of the triangles
A
B
C
ABC
A
BC
and
A
P
D
APD
A
P
D
. Prove that
∠
B
A
Q
=
∠
C
A
J
\angle BAQ=\angle CAJ
∠
B
A
Q
=
∠
C
A
J
.
geometry
incenter