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

Author

Michel Lagneau, Dec 07 2013

Keywords

• nonn

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

• Eric Weisstein's World of Mathematics, Excircles.
• Eric Weisstein's World of Mathematics, Exradius.
• Eric Weisstein's World of Mathematics, Inradius.

Crossrefs

Cf. A185210.

Programs

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]