OFFSET
1,1
COMMENTS
Descartes's theorem: 4 kissing circles with radii a,b,c,d satisfy
(1/a + 1/b + 1/c + 1/d)^2 = 2 (1/a^2 + 1/b^2 + 1/c^2 + 1/d^2).
When the largest circle encloses other 3 circles, its radius is negative.
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.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..837
Eric Weisstein's World of Mathematics, Soddy Circles.
Eric Weisstein's World of Mathematics, Tangent Circles
Wikipedia, Descartes' theorem
EXAMPLE
The table gives the first 8 examples:
a b c d
== == == ==
6 3 2 1
6 3 3 2
15 10 3 2
18 9 8 8
20 12 5 3
21 14 7 6
28 21 4 3
35 15 14 6
MATHEMATICA
aMax = 150; (* WARNING: O(n^3) *)
Do[
If[x \[NotElement] Rationals, Continue[]];
{d1, d2} = 1/(-(1/a) + 1/b + 1/c + {1, -1} 2 x);
If[GCD[a, b, c, d1] == 1, {a, b, c, d1} // Sow];
If[d2 > c || d2 == d1, Continue[]];
If[GCD[a, b, c, d2] == 1, {a, b, c, d2} // Sow];
, {a, aMax}, {b, 2, a - 2}, {c, Min[b, a - b]}
, {x, {Sqrt[(-a + b + c)/(-a b c)]}}] // Reap // Last // Last // TableForm
CROSSREFS
KEYWORD
nonn
AUTHOR
Albert Lau, Aug 13 2017
EXTENSIONS
Description clarified by Ray Chandler, Aug 19 2017
STATUS
approved