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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2003 National Olympiad First Round
21
21
Part of
2003 National Olympiad First Round
Problems
(1)
P21 [Geometry] - Turkish NMO 1st Round - 2003
Source:
5/16/2014
The circle
C
1
C_1
C
1
and
C
2
C_2
C
2
are externally tangent to each other at
T
T
T
. A line passing through
T
T
T
meets
C
1
C_1
C
1
at
A
A
A
and meets
C
2
C_2
C
2
at
B
B
B
. The line which is tangent to
C
1
C_1
C
1
at
A
A
A
meets
C
2
C_2
C
2
at
D
D
D
and
E
E
E
. If
D
∈
[
A
E
]
D \in [AE]
D
∈
[
A
E
]
,
∣
T
A
∣
=
a
|TA|=a
∣
T
A
∣
=
a
,
∣
T
B
∣
=
b
|TB|=b
∣
TB
∣
=
b
, what is
∣
B
E
∣
|BE|
∣
BE
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
a
(
a
+
b
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
a
2
+
b
2
+
a
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
a
2
+
b
2
−
a
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
a
2
+
b
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
(
a
+
b
)
b
<span class='latex-bold'>(A)</span>\ \sqrt{a(a+b)} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{a^2+b^2+ab} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{a^2+b^2-ab} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{a^2+b^2} \qquad<span class='latex-bold'>(E)</span>\ \sqrt{(a+b)b}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
a
(
a
+
b
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
a
2
+
b
2
+
ab
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
a
2
+
b
2
−
ab
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
a
2
+
b
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
(
a
+
b
)
b
geometry