A circle centered at P with radius 4 cm and another centered at Q with radius 16 cm touch each other externally. A third circle with centre R is drawn to touch the first two circles. they all have a common tangent. Then the radius of the circle with centre R is

i am giving more possibilities than given earlier.... there are two circles external to the given circles... then there are infinite circles that are enclosed with in the given two circles. Also there are infinite circles with radius R which are enclosing the given two circles.



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There are infinite circles with center R and with any radius.
See diagram.

Given two circles are 5 and 6.

The blue circles  numbered 1 to 4 are all possible solutions.  These are infinite in number.  They are within circle 5 or circle 6.  Also they are outside circle 5 and outside circle 6.

The centers of all circles  ie., P, Q and R are collinear.

obviously the common tangent is perpendicular to the line joining all centers of all circles.

Also you find two more circles  numbered 7 and 8.,  These are tangentially touching circles 5 and 6, as well as the common tangent of circles 5 and 6.

1 5 1
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