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A210207 Area A of the non-right triangles such that A, the sides, and the circumradius are integers.

%I A210207 #19 Feb 16 2025 08:33:17
%S A210207 168,432,480,624,672,768,1320,1512,1536,1560,1680,1728,1848,1920,2040,
%T A210207 2304,2376,2496,2520,2688,2856,3024,3072,3240,3696,3720,3840,3864,
%U A210207 3888,4104,4200,4320,4536,5280,5376,5616,5712,6000,6048,6144,6240,6552,6720,6912
%N A210207 Area A of the non-right triangles such that A, the sides, and the circumradius are integers.
%C A210207 A103251 gives the areas of right triangles with the same property (the area, the sides, and the circumradius are integers). Thus the intersection of this sequence with A103251 will give the areas of 2 families of triangles with the same property: one family of right triangles and one family of non-right triangles.
%C A210207 For example a(3) = A103251(8) = 480 generates two triangles whose sides are
%C A210207 (a,b,c) = (32, 50, 78) = > A = 480, R = 65, and 32^2 + 50^2 is no square;
%C A210207 (a,b,c) = (20, 48, 52) = > A = 480, R = 26, and 20^2 + 48^2  = 52^2 is square.
%C A210207 {a(n) intersection A103251} = {480, 1320, 1536, 1920, 2520, 3024, 3696, 3840, ...}
%H A210207 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 A210207 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Circumradius.html">MathWorld: Circumradius</a>
%F A210207 Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);
%F A210207 Circumradius R = a*b*c/4A.
%e A210207 168 is in the sequence because, for (a,b,c) = (14,30,40), A = sqrt(42*(42-14)*(42-30)*(42-40)) = 168, and 14^2 + 30^2 is no square.
%p A210207 T:=array(1..4000):nn:=400:k:=0:for a from 1
%p A210207 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): u:=a^2+b^2:if  x>0 then x1:=sqrt(x) : y:=a*b*c/(4*x1):
%p A210207 else fi:if x1=floor(x1) and y = floor(y) and u <> c^2 then k:=k+1:T[k]:=x1:else fi:od:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):
%t A210207 nn=400; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0 < area2 && a^2 != b^2+c^2 && IntegerQ[Sqrt[area2]] && IntegerQ[a*b*c/(4*Sqrt[area2])], AppendTo[lst, Sqrt[area2]]]], {a,nn}, {b,a}, {c,b}]; Union[lst]
%Y A210207 Cf. A103251, A208984, A188158.
%K A210207 nonn
%O A210207 1,1
%A A210207 _Michel Lagneau_, Mar 18 2012