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Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2003 National Olympiad First Round
33
33
Part of
2003 National Olympiad First Round
Problems
(1)
P33 [Geometry] - Turkish NMO 1st Round - 2003
Source:
5/31/2014
Let
G
G
G
be the intersection of medians of
△
A
B
C
\triangle ABC
△
A
BC
and
I
I
I
be the incenter of
△
A
B
C
\triangle ABC
△
A
BC
. If
∣
A
B
∣
=
c
|AB|=c
∣
A
B
∣
=
c
,
∣
A
C
∣
=
b
|AC|=b
∣
A
C
∣
=
b
and
G
I
⊥
B
C
GI \perp BC
G
I
⊥
BC
, what is
∣
B
C
∣
|BC|
∣
BC
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
b
+
c
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
b
+
c
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
b
2
+
c
2
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
b
2
+
c
2
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ \dfrac{b+c}2 \qquad<span class='latex-bold'>(B)</span>\ \dfrac{b+c}{3} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{\sqrt{b^2+c^2}}{2} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{\sqrt{b^2+c^2}}{3\sqrt 2} \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
b
+
c
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
b
+
c
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
b
2
+
c
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
2
b
2
+
c
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
geometry
incenter