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2022 Taiwan Mathematics Olympiad
5
5
Part of
2022 Taiwan Mathematics Olympiad
Problems
(1)
2022 TMO Geometry
Source: 2022 Taiwan Mathematics Olympiad
2/19/2022
Let
J
J
J
be the
A
A
A
-excenter of an acute triangle
A
B
C
ABC
A
BC
. Let
X
X
X
,
Y
Y
Y
be two points on the circumcircle of the triangle
A
C
J
ACJ
A
C
J
such that
J
X
‾
=
J
Y
‾
<
J
C
‾
\overline{JX} = \overline{JY} < \overline{JC}
J
X
=
J
Y
<
J
C
. Let
P
P
P
be a point lies on
X
Y
XY
X
Y
such that
P
B
PB
PB
is tangent to the circumcircle of the triangle
A
B
C
ABC
A
BC
. Let
Q
Q
Q
be a point lies on the circumcircle of the triangle
B
X
Y
BXY
BX
Y
such that
B
Q
BQ
BQ
is parallel to
A
C
AC
A
C
. Prove that
∠
B
A
P
=
∠
Q
A
C
\angle BAP = \angle QAC
∠
B
A
P
=
∠
Q
A
C
.Proposed by Li4.
geometry