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A290931 Radius of a circle enclosing three mutually tangent circles, such that they have coprime integer radii and with collectively six distinct points of tangency.

%I A290931 #27 Feb 16 2025 08:33:50
%S A290931 6,6,15,18,20,21,28,35,40,40,42,42,45,45,52,54,56,63,66,66,72,75,77,
%T A290931 88,91,95,99,100,104,105,105,110,112,117,120,126,130,143,153,153,156,
%U A290931 160,160,165,165,168,170,175,186,187,189,190,195,196,198,198,204,208,208
%N A290931 Radius of a circle enclosing three mutually tangent circles, such that they have coprime integer radii and with collectively six distinct points of tangency.
%C A290931 Descartes's theorem: 4 kissing circles with radii a,b,c,d satisfy
%C A290931 (1/a + 1/b + 1/c + 1/d)^2 = 2 (1/a^2 + 1/b^2 + 1/c^2 + 1/d^2).
%C A290931 When the largest circle encloses other 3 circles, its radius is negative.
%C A290931 If all circles are tangent to each other at the same point, Descartes's theorem does not apply. In this case, all circles can have any radius.
%H A290931 Ray Chandler, <a href="/A290931/b290931.txt">Table of n, a(n) for n = 1..837</a>
%H A290931 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SoddyCircles.html">Soddy Circles.</a>
%H A290931 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TangentCircles.html">Tangent Circles</a>
%H A290931 Wikipedia, <a href="https://en.wikipedia.org/wiki/Descartes%27_theorem">Descartes' theorem</a>
%e A290931 The table gives the first 8 examples:
%e A290931    a  b  c  d
%e A290931   == == == ==
%e A290931    6  3  2  1
%e A290931    6  3  3  2
%e A290931   15 10  3  2
%e A290931   18  9  8  8
%e A290931   20 12  5  3
%e A290931   21 14  7  6
%e A290931   28 21  4  3
%e A290931   35 15 14  6
%t A290931 aMax = 150; (* WARNING: O(n^3) *)
%t A290931 Do[
%t A290931      If[x \[NotElement] Rationals, Continue[]];
%t A290931      {d1, d2} = 1/(-(1/a) + 1/b + 1/c + {1, -1} 2 x);
%t A290931      If[GCD[a, b, c, d1] == 1, {a, b, c, d1} // Sow];
%t A290931      If[d2 > c || d2 == d1, Continue[]];
%t A290931      If[GCD[a, b, c, d2] == 1, {a, b, c, d2} // Sow];
%t A290931      , {a, aMax}, {b, 2, a - 2}, {c, Min[b, a - b]}
%t A290931      , {x, {Sqrt[(-a + b + c)/(-a b c)]}}] // Reap // Last // Last // TableForm
%Y A290931 Cf. A290508 (4 circles tangent externally).
%K A290931 nonn
%O A290931 1,1
%A A290931 _Albert Lau_, Aug 13 2017
%E A290931 Description clarified by _Ray Chandler_, Aug 19 2017