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Baltic Way
1991 Baltic Way
16
Equation between three tangent circles
Equation between three tangent circles
Source:
April 19, 2013
Problem Statement
Two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
with radii
r
1
r_1
r
1
and
r
2
r_2
r
2
touch each other externally and both touch a line
l
l
l
. A circle
C
3
C_3
C
3
with radius
r
3
<
r
1
,
r
2
r_3 < r_1, r_2
r
3
<
r
1
,
r
2
is tangent to
l
l
l
and externally to
C
1
C_1
C
1
and
C
2
C_2
C
2
. Prove that
1
r
3
=
1
r
2
+
1
r
2
.
\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_2}}+\frac{1}{\sqrt{r_2}}.
r
3
1
=
r
2
1
+
r
2
1
.
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