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IberoAmerican
2021 Iberoamerican
2
2
Part of
2021 Iberoamerican
Problems
(1)
Midpoints and circumcircles lead to another midpoint
Source: 2021 Iberoamerican Mathematical Olympiad, P2
10/20/2021
Consider an acute-angled triangle
A
B
C
ABC
A
BC
, with
A
C
>
A
B
AC>AB
A
C
>
A
B
, and let
Γ
\Gamma
Γ
be its circumcircle. Let
E
E
E
and
F
F
F
be the midpoints of the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. The circumcircle of the triangle
C
E
F
CEF
CEF
and
Γ
\Gamma
Γ
meet at
X
X
X
and
C
C
C
, with
X
≠
C
X\neq C
X
=
C
. The line
B
X
BX
BX
and the tangent to
Γ
\Gamma
Γ
through
A
A
A
meet at
Y
Y
Y
. Let
P
P
P
be the point on segment
A
B
AB
A
B
so that
Y
P
=
Y
A
YP = YA
Y
P
=
Y
A
, with
P
≠
A
P\neq A
P
=
A
, and let
Q
Q
Q
be the point where
A
B
AB
A
B
and the parallel to
B
C
BC
BC
through
Y
Y
Y
meet each other. Show that
F
F
F
is the midpoint of
P
Q
PQ
PQ
.
geometry
circumcircle