Three circles of radii touch each other externally. If they have -axis as a common tangent, then:

Question

Three circles of radii  touch each other externally. If they have -axis as a common tangent, then:

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🧠 Reading the Question — Standard Descartes-Style Identity for Three Mutually-Tangent Circles on a Line

Three circles touching each other externally and all touching the -axis: each has its centre at height  its radius above the -axis. So centres are , ,  for the three circles of radii .

External tangency between two such circles (radii ):

So .

Three circles, smallest in the middle: since , the smallest circle (radius ) sits between the other two on the -axis. Let's say (without loss

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