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IberoAmerican
2020 IberoAmerican
6
6
Part of
2020 IberoAmerican
Problems
(1)
Circles and the Euler line
Source: 2020 Iberoamerican #6
11/17/2020
Let
A
B
C
ABC
A
BC
be an acute, scalene triangle. Let
H
H
H
be the orthocenter and
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
, and let
P
P
P
be a point interior to the segment
H
O
.
HO.
H
O
.
The circle with center
P
P
P
and radius
P
A
PA
P
A
intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
again at
R
R
R
and
S
S
S
, respectively. Denote by
Q
Q
Q
the symmetric point of
P
P
P
with respect to the perpendicular bisector of
B
C
BC
BC
. Prove that points
P
P
P
,
Q
Q
Q
,
R
R
R
and
S
S
S
lie on the same circle.
geometry