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Question from Kevin, a teacher:

3 given circles of R80, R56 & R24 are all in contact. The 2 smaller ones are inscribed in the big one.
Find by calculation or graphically (both if possible) the radiusof a 3rd circle which will be in contact with all 3 given circles.

Kevin,

The answer is given by the Descartes Circle Theorem:

In the formula there you must take the radius of the big (outer) circle to be NEGATIVE! Note that the numbers in the formula are reciprocals of the given radii -- k = 1/r.

The formula, for your numbers, says that
2[80^-2 + 56^-2 + 24^-2 + x^-2] = [(-80)^-1 + 56^-1 + 24^-1 + x^-1]².

Now, solve the resulting quadratic equation for x.

Chris

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