A231174 Integer areas of integer-sided triangles such that the distance between the incenter and the circumcenter is an integer.

48, 192, 432, 768, 1200, 1728, 2352, 3072, 3840, 3888, 4800, 5808, 6000, 6912, 8112, 9408, 10800, 12288, 12960, 13872, 15360, 15552, 17328, 19200, 21168, 23232, 24000, 25392, 26880, 27648, 30000, 30720, 32448, 32928, 34560, 34992, 37632, 40368, 43200, 46128

Offset: 1

Author

Michel Lagneau, Nov 05 2013

Keywords

• nonn

Comments

The distance between the incenter and circumcenter is given by d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius, a result known as the Euler triangle formula (see the link below).

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.

The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.

Properties of this sequence:

It appears that the triangles are isosceles.

a(n) = 48*m where the integers m are not squarefree: {m} ={1, 4, 9, 16, 25, 36, 49, 64, 80, 81, 100, 121, 125, 144, 169, 196, 225, 256, 270, 289, ...}, and the areas of the primitive triangles are 48, 3840, 6000, ... The integers m are not squarefree.

The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.

The subsequence of the areas of triangles with inradius, circumradius and distance between the incenter and circumcenter integers is {432, 1728, 3072, 3888, 6000, 6912, ...}.

The following table gives the first values (A, a, b, c, R, r, d) where A is the area of the triangles, a, b, c are the integer sides of the triangles, R is the circumradius, r is the inradius and d is the distance between the incenter and circumcenter:

--------------------------------------------

| A | a | b | c | R | r | d |

--------------------------------------------

| 48 | 10 | 10 | 16 | 25/3 | 8/3 | 5 |

| 192 | 20 | 20 | 32 | 50/3 | 16/3 | 10 |

| 432 | 30 | 30 | 48 | 25 | 8 | 15 |

| 768 | 40 | 40 | 64 | 100/3| 32/3 | 20 |

| 1200 | 50 | 50 | 80 | 125/3| 40/3 | 25 |

| 1728 | 60 | 60 | 96 | 50 | 16 | 30 |

| 2352 | 70 | 70 | 112 | 175/3| 56/3 | 35 |

| 3072 | 80 | 80 | 96 | 50 | 24 | 10 |

| 3072 | 80 | 80 | 128 | 200/3| 64/3 | 40 |

| 3840 | 80 | 104 | 104 | 169/3| 80/3 | 13 |

| 3888 | 90 | 90 | 144 | 75 | 24 | 45 |

| 4800 |100 | 100 | 160 | 250/3| 80/3 | 50 |

| 5808 |110 | 110 | 176 | 275/3| 88/3 | 55 |

| 6000 |130 | 130 | 240 | 169 | 24 |143 |

| 6912 |120 | 120 | 144 | 75 | 36 | 15 |

| 6912 |120 | 120 | 192 | 100 | 32 | 60 |

..........................................

Links

• Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
• Eric Weisstein's World of Mathematics, Circumcenter.
• Eric Weisstein's World of Mathematics, Incenter.

Crossrefs

Cf. A188158.

Programs

Mathematica

nn=800;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[a*b*c/(4*Sqrt[area2])*(a*b*c/(4*Sqrt[area2])-2*Sqrt[area2]/s)]],AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]