Kissing Circles (Descartes Theorem)

Geometry Level 4

Three circles of radii 4, 7, 10 are positioned such that they are mutually tangent to each other (kissing). Two additional circles can be drawn such that they are tangent to all of these three circles. A smaller circle of radius \( r \) that lies in between the three circles and a larger circle of radius \( R \) that contains all of the three circles and is tangent to them on the inside.

Use Descartes' circle theorem to calculate the radii of these two additional circles.

Submit as your answer the sum \( r + R \).

Give your answer to 2 decimal places.


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