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Middle European Mathematical Olympiad
2023 Middle European Mathematical Olympiad
6
6
Part of
2023 Middle European Mathematical Olympiad
Problems
(1)
Hard excircle geo
Source: MEMO 2023 T6
8/25/2023
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
. Let
J
J
J
be the center of the
A
A
A
-excircle of
A
B
C
ABC
A
BC
. Let
D
D
D
be the projection of
J
J
J
on line
B
C
BC
BC
. The internal bisectors of angles
B
D
J
BDJ
B
D
J
and
J
D
C
JDC
J
D
C
intersectlines
B
J
BJ
B
J
and
J
C
JC
J
C
at
X
X
X
and
Y
Y
Y
, respectively. Segments
X
Y
XY
X
Y
and
J
D
JD
J
D
intersect at
P
P
P
. Let
Q
Q
Q
be the projection of
A
A
A
on line
B
C
BC
BC
. Prove that the internal angle bisector of
Q
A
P
QAP
Q
A
P
is perpendicular to line
X
Y
XY
X
Y
.Proposed by Dominik Burek, Poland
geometry