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2022 Nordic
4
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Part of
2022 Nordic
Problems
(1)
Nice geometry:three circles concur
Source: 2022 Nordic Mathematical Olympiad
4/16/2022
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with circumscribed circle
k
k
k
and centre of the circumscribed circle
O
O
O
. A line through
O
O
O
intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
.Denote by
B
′
B'
B
′
and
C
′
C'
C
′
the reflections of
B
B
B
and
C
C
C
over
O
O
O
, respectively. Prove that the circumscribed circles of
O
D
C
′
ODC'
O
D
C
′
and
O
E
B
′
OEB'
OE
B
′
concur on
k
k
k
.
geometry
circumcircle