Two circles with radius a and b touch each other externally....... And son socratic
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Let
A
,
B
,
C
be the center of circle 1, circle 2 and circle 3, respectively, as shown in the figure.
A
B
=
a
+
b
,
A
C
=
a
+
c
,
B
C
=
b
+
c
In
Δ
C
D
B
,
C
D
2
=
B
C
2
−
B
D
2
⇒
C
D
2
=
(
b
+
c
)
2
−
(
b
−
c
)
2
⇒
C
D
2
=
(
b
+
c
+
b
−
c
)
⋅
(
b
+
c
−
b
+
c
)
=
2
b
⋅
2
c
=
4
b
c
⇒
C
D
=
√
4
b
c
=
2
√
b
c
Similarly,
C
F
2
=
(
a
+
c
)
2
−
(
a
−
c
)
2
⇒
C
F
=
2
√
a
c
⇒
D
F
=
C
F
+
C
D
=
2
√
a
c
+
2
√
b
c
=
2
√
c
(
√
a
+
√
b
)
In
Δ
A
E
B
,
B
E
2
=
A
B
2
−
A
E
2
⇒
B
E
2
=
(
a
+
b
)
2
−
(
b
−
a
)
2
⇒
B
E
2
=
(
a
+
b
+
b
−
a
)
⋅
(
a
+
b
−
b
+
a
)
=
2
b
⋅
2
a
=
4
a
b
⇒
B
E
=
√
4
a
b
=
2
√
a
b
As
B
E
=
D
F
,
⇒
2
√
a
b
=
2
√
c
(
√
a
+
√
b
)
⇒
1
√
c
=
√
a
+
√
b
√
a
b
⇒
1
√
c
=
√
a
√
a
b
+
√
b
√
a
b
⇒
1
√
c
=
1
√
b
+
1
√
a
⇒
1
√
c
=
1
√
a
+
1
√
b
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