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Turkey Team Selection Test
2017 Turkey Team Selection Test
8
8
Part of
2017 Turkey Team Selection Test
Problems
(1)
Turkey TST 2017 P8
Source: Turkey TST 2017 P8
6/1/2017
In a triangle
A
B
C
ABC
A
BC
the bisectors through vertices
B
B
B
and
C
C
C
meet the sides
[
A
C
]
\left [ AC \right ]
[
A
C
]
and
[
A
B
]
\left [ AB \right ]
[
A
B
]
at
D
D
D
and
E
E
E
respectively. Let
I
c
I_{c}
I
c
be the center of the excircle which is tangent to the side
[
A
B
]
\left [ AB \right ]
[
A
B
]
and
F
F
F
the midpoint of
[
B
I
c
]
\left [ BI_{c} \right ]
[
B
I
c
]
. If
∣
C
F
∣
2
=
∣
C
E
∣
2
+
∣
D
F
∣
2
\left | CF \right |^2=\left | CE \right |^2+\left | DF \right |^2
∣
CF
∣
2
=
∣
CE
∣
2
+
∣
D
F
∣
2
, show that
A
B
C
ABC
A
BC
is an equilateral triangle.
Turkey
TST
geometry