Circles of Apollonius

radii:

blue circle

5

red circle

10

yellow circle

6

outside tangent:

blue circle

red circle

yellow circle

radius of new circle is:

3.89052

Apollonius' problem asks for the circle tangent to three given circles or lines. Finding a circle tangent to three circles is the most difficult form of this problem. The solution is obtained by solving three simultaneous equations,

2

(x-

x

1

)

+

2

(y-

y

1

)

-

2

(r±

r

1

)

=0

2

(x-

x

2

)

+

2

(y-

y

2

)

-

2

(r±

r

2

)

=0

2

(x-

x

3

)

+

2

(y-

y

3

)

-

2

(r±

r

3

)

=0

,

with at most eight solutions.