A231174 - OEIS

%I #16 Feb 24 2023 16:29:15

%S 48,192,432,768,1200,1728,2352,3072,3840,3888,4800,5808,6000,6912,

%T 8112,9408,10800,12288,12960,13872,15360,15552,17328,19200,21168,

%U 23232,24000,25392,26880,27648,30000,30720,32448,32928,34560,34992,37632,40368,43200,46128

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

%C 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).

%C 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.

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

%C Properties of this sequence:

%C It appears that the triangles are isosceles.

%C 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.

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

%C 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, ...}.

%C 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:

%C --------------------------------------------

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

%C --------------------------------------------

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

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

%C |  432 | 30 |  30 |  48 | 25   |  8   | 15 |

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

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

%C | 1728 | 60 |  60 |  96 |  50  | 16   | 30 |

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

%C | 3072 | 80 |  80 |  96 |  50  | 24   | 10 |

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

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

%C | 3888 | 90 |  90 | 144 |  75  | 24   | 45 |

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

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

%C | 6000 |130 | 130 | 240 | 169  | 24   |143 |

%C | 6912 |120 | 120 | 144 |  75  | 36   | 15 |

%C | 6912 |120 | 120 | 192 | 100  | 32   | 60 |

%C ..........................................

%H Mohammad K. Azarian, <a href="http://www.jstor.org/stable/25678790">Solution to Problem S125: Circumradius and Inradius</a>, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Circumcenter.html">Circumcenter</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incenter.html">Incenter</a>.

%t 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]

%Y Cf. A188158.

%K nonn

%O 1,1

%A _Michel Lagneau_, Nov 05 2013