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Contests
National and Regional Contests
Japan Contests
Japan TST
2023 Japan TST
11
11
Part of
2023 Japan TST
Problems
(1)
Japanese Geo
Source: 2023 Japan TST p11
7/21/2023
Let
A
B
C
ABC
A
BC
be an acute triangle and point
P
P
P
lies inside the triangle (excluding the vertices on the boundary) such that lines
A
P
AP
A
P
and
B
C
BC
BC
are not orthogonal. Let
X
X
X
and
Y
Y
Y
be the points symmetric to
P
P
P
wrt lines
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, and let
ω
\omega
ω
be the circumcircle of triangle
A
X
Y
AXY
A
X
Y
. Point
Q
Q
Q
lies inside triangle
A
B
C
ABC
A
BC
(excluding the vertices on the boundary) and satisfies
∠
Q
B
C
=
∠
C
A
P
\angle QBC = \angle CAP
∠
QBC
=
∠
C
A
P
,
∠
Q
C
B
=
∠
B
A
P
\angle QCB = \angle BAP
∠
QCB
=
∠
B
A
P
. Line
A
Q
AQ
A
Q
intersects
ω
\omega
ω
at a point
R
R
R
, distinct from
A
A
A
and
Q
Q
Q
. Prove that the circumcircles of triangles
A
B
C
ABC
A
BC
,
P
Q
R
PQR
PQR
, and
ω
\omega
ω
have a common point.
geometry