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Given three circles with diameters of .623", .687", and .719" that fit snugly within a circle of diameter D, what is D? What is the mathematical formula for this?

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    $\begingroup$ looking on how to find the diameter of the larger circle by using the known diameter of three smaller circles of slightly different diameters that fit within this circle. ie checking the diameter of a given hole using three gauge pins of different known diameters. gauge pins all have three points of contact/touching within this unknown diameter. $\endgroup$ Commented 2 days ago
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    $\begingroup$ en.wikipedia.org/wiki/Descartes%27_theorem and the formula using radii at en.wikipedia.org/wiki/Apollonian_gasket#Construction where you would take $r_4$ as negative because you are asking about the surrounding circle $\endgroup$ Commented 2 days ago
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    $\begingroup$ "define this term FIT SNUGLY,if their origins lie on the same straight line" I don't think ANYONE would define "fit snugly" that way. Might as well define fit snugly as be concentric and take the max diameter. I agree that it is beholden on the OP to describe the problem clearly and ambiguously but I think we forget how difficult that is for beginners who just don't have the vocabulary and experience for that. Even I'm having a hard time putting into words which I think it is "obvious" they mean: three pairwise tangential non-intersecting circles and smallest fourth circle containing them $\endgroup$ Commented 2 days ago
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    $\begingroup$ @WillJagy I didn't even notice that word transposition. I am assuming they OP is talking of the "kissing circles" of your links and that your links answer the question fully (post as answer?). I feel we do a lot of: OP posts something with poor wording, posters respond with "you are being vague-- its your fault we don't understand you. Voting to close". Which is... fair I guess when the question really is vague. But I think we should have a little compassion and remember how difficult and subtle it can be to find the right words to describe a concept one's only minimal experienced in. $\endgroup$ Commented 2 days ago
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    $\begingroup$ If $r_4$ is the radius of the largest, and surrounding, circle, we may force $r_4 > 0$ by writing $$ \left( \frac{1}{r_1 } + \frac{1}{r_2 } + \frac{1}{r_3 } - \frac{1}{r_4 } \right)^2 = 2 \left( \frac{1}{r_1^2 } + \frac{1}{r_2^2 } + \frac{1}{r_3^2 } + \frac{1}{r_4^2 } \right) $$ $\endgroup$ Commented 2 days ago

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