Math CentralQuandaries & Queries


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.


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.


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