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A208984 Areas A of the triangles such that A, the sides, the circumradius and the inradius are integers.

%I A208984 #19 Feb 16 2025 08:33:16
%S A208984 24,96,120,168,216,240,336,384,432,480,600,624,672,720,768,840,864,
%T A208984 960,1080,1176,1320,1344,1512,1536,1560,1680,1728,1848,1920,1944,2016,
%U A208984 2040,2160,2184,2304,2376,2400,2496,2520,2688,2856,2880,2904,3000,3024,3072,3240
%N A208984 Areas A of the triangles such that A, the sides, the circumradius and the inradius are integers.
%C A208984 a(n) is divisible by 24, and the positive squares A000290(n) are included in the sequence a(n)/24 = {1, 4, 5, 7, 9, 10, 14, 16, 18, 20, 25, 26, 28, 30, 32, 35, 36, 40, 45, 49, 55, 56, 63, 64, 65, ...}.
%C A208984 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.
%H A208984 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/25678790">Solution of problem 125: Circumradius and Inradius</a>, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
%H A208984 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Circumradius.html">MathWorld: Circumradius</a>
%H A208984 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Inradius.html">MathWorld: Inradius</a>
%e A208984 a(1) = 24 because, for (a,b,c) = (6, 8, 10) => s = (6 + 8 + 10)/2 = 12, and
%e A208984 A = sqrt(12(12-6)(12-8)(12-10)) = sqrt(576) = 24;
%e A208984 R = abc/4A = 480/4*24 = 5;
%e A208984 r = A/p = 24/12 = 2.
%p A208984 with(numtheory):T:=array(1..1000):k:=0:nn:=250: for a from 1
%p A208984 to nn do: for b from a to nn  do: for c from b to nn  do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then s:=sqrt(x) :if s=floor(s) and irem(a*b*c,4*s) = 0 and irem(s,p)=0 then k:=k+1:T[k]:= s: else fi:fi:od:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):
%t A208984 nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]] && IntegerQ[a*b*c/(4* Sqrt[area2])] && IntegerQ[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y A208984 Cf. A120062, A188158.
%K A208984 nonn
%O A208984 1,1
%A A208984 _Michel Lagneau_, Mar 04 2012