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Turkey MO (2nd round)
2019 Turkey MO (2nd round)
4
4
Part of
2019 Turkey MO (2nd round)
Problems
(1)
Proving the equilateral triangle
Source: Turkey National Mathematical Olympiad 2019 P4
12/23/2019
In a triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
,
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
. Let
M
M
M
be on the minor arc
A
C
AC
A
C
of the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
different than
A
A
A
and
C
C
C
. Let
B
M
BM
BM
and
A
C
AC
A
C
meet at
E
E
E
and the bisector of
∠
B
M
C
\angle BMC
∠
BMC
and
B
C
BC
BC
meet at
F
F
F
such that
∠
A
F
B
=
∠
C
F
E
\angle AFB=\angle CFE
∠
A
FB
=
∠
CFE
. Prove that the triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
is equilateral.
geometry