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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.
2
6, 6, 15, 18, 20, 21, 28, 35, 40, 40, 42, 42, 45, 45, 52, 54, 56, 63, 66, 66, 72, 75, 77, 88, 91, 95, 99, 100, 104, 105, 105, 110, 112, 117, 120, 126, 130, 143, 153, 153, 156, 160, 160, 165, 165, 168, 170, 175, 186, 187, 189, 190, 195, 196, 198, 198, 204, 208, 208
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
Eric Weisstein's World of Mathematics, Soddy Circles.
Eric Weisstein's World of Mathematics, Tangent Circles
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
Cf. A290508 (4 circles tangent externally).
Sequence in context: A266223 A256675 A362534 * A257372 A058563 A175561
KEYWORD
nonn
AUTHOR
Albert Lau, Aug 13 2017
EXTENSIONS
Description clarified by Ray Chandler, Aug 19 2017
STATUS
approved