A233317 Integer areas A of the integer-sided triangles such that the inradius and the radius of the three excircles are perfect squares.

108, 1728, 8748, 12348, 27648, 67500, 139968, 197568, 259308, 442368, 707472, 708588, 1000188, 1080000, 1581228, 2239488, 3084588, 3161088, 4148928, 5467500, 7077888, 7717500, 9020268, 11319552, 11337408, 14074668, 16003008, 17280000, 21003948, 25299648

Offset

1,1

Comments

Subset of A185210.

A = sqrt(s*(p-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);

The inradius is r = A/s;

The radii of the three excircles are r1 = 2*A/(-a+b+c); x2 = 2*A*b/(a-b+c); x3 = 2*A*c/(a+b-c).

The areas A of the primitive triangles of sides (a,b,c) are 108, 12348, ...

The areas of the nonprimitive triangles of sides (a*p^2, b*p^2, c*p^2) are in the sequence with the value A*p^4.

The following table gives the first values (A, a, b, c, r, r1, r2, r3) where A is the area of the triangle, a, b, c the integer sides, r, r1, r2 and r3 are respectively the length of the inradius and the radius of the three excircles.

+--------+-----+------+------+------+------+------+------+

| A | a | b | c | r | r1 | r2 | r3 |

+--------+-----+------+------+------+------+------+------+

| 108 | 15 | 15 | 24 | 2^2 | 3^2 | 3^2 | 6^2 |

| 1728 | 60 | 60 | 96 | 4^2 | 6^2 | 6^2 | 12^2 |

| 8748 | 135 | 135 | 216 | 6^2 | 9^2 | 9^2 | 18^2 |

| 12348 | 91 | 280 | 315 | 6^2 | 7^2 | 14^2 | 21^2 |

| 27648 | 240 | 240 | 384 | 8^2 | 12^2 | 12^2 | 24^2 |

| 67500 | 375 | 375 | 600 | 10^2 | 15^2 | 15^2 | 30^2 |

| 139968 | 540 | 540 | 864 | 12^2 | 18^2 | 18^2 | 36^2 |

| 197568 | 364 | 1120 | 1260 | 12^2 | 14^2 | 28^2 | 42^2 |

| 259308 | 735 | 735 | 1176 | 14^2 | 21^2 | 21^2 | 42^2 |

+--------+-----+------+------+------+------+------+------+

Links

Table of n, a(n) for n=1..30.

Eric Weisstein's World of Mathematics, Excircles.

Eric Weisstein's World of Mathematics, Exradius.

Eric Weisstein's World of Mathematics, Inradius.

Mathematica

nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[area2>0&& IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[Sqrt[area2]/s]] && IntegerQ[Sqrt[2*Sqrt[area2]/(-a+b+c)]] && IntegerQ[Sqrt[2*Sqrt[area2]/(a-b+c)]] && IntegerQ[Sqrt[2*Sqrt[area2]/(a+b-c)]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Crossrefs

Cf. A185210.

Sequence in context: A115135 A202309 A279981 * A263970 A250339 A263961

Adjacent sequences: A233314 A233315 A233316 * A233318 A233319 A233320

Keyword

nonn

Author

Michel Lagneau, Dec 07 2013

Status

approved