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Balkan MO Shortlist
2023 Balkan MO Shortlist
G2
G2
Part of
2023 Balkan MO Shortlist
Problems
(1)
Similar triangles and equal segments
Source: BMO SL 2023 G2
5/3/2024
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with circumcenter
O
O
O
lying in the interior. Let
E
E
E
and
F
F
F
be the midpoints of the segments
B
C
BC
BC
and
A
D
AD
A
D
, respectively. Let
X
X
X
be the point lying on the same side of the line
E
F
EF
EF
as the vertex
C
C
C
such that
△
E
X
F
\triangle EXF
△
EXF
and
△
B
O
A
\triangle BOA
△
BO
A
are similar. Prove that
X
C
=
X
D
XC = XD
XC
=
X
D
.
geometry
BMO Shortlist