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A232274 Integer areas A of the integer-sided triangles such that the length of the inradius and the circumradius are both a perfect square.

%I A232274 #22 Feb 16 2025 08:33:20
%S A232274 168,2688,13608,43008,105000,108000,217728,403368,688128,1102248,
%T A232274 1680000,1728000
%N A232274 Integer areas A of the integer-sided triangles such that the length of the inradius and the circumradius are both a perfect square.
%C A232274 Subset of A208984.
%C A232274 The areas of the primitive triangles of sides (a, b, c) and inradius, circumradius equals respectively to r and R are 672, 108000, ...  The sides of the nonprimitive triangles are of the form (a*k^2, b*k^2, c*k^2) with r' = r*k^2 and R' = R*k^2 where r', R' are respectively the inradius and the circumradius of the nonprimitive triangles. The areas A' of the nonprimitive triangles are A' = A*k^4.
%C A232274 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.
%C A232274 The following table gives the first values (A, a, b, c, r, R).
%C A232274 +---------+------+------+------+-----+------+
%C A232274 |    A    |   a  |   b  |   c  |  r  |   R  |
%C A232274 +---------+------+------+------+-----+------+
%C A232274 |     168 |   14 |   30 |   40 |   4 |   25 |
%C A232274 |    2688 |   56 |  120 |  160 |  16 |  100 |
%C A232274 |   13608 |  126 |  270 |  360 |  36 |  225 |
%C A232274 |   43008 |  224 |  480 |  640 |  64 |  400 |
%C A232274 |  105000 |  350 |  750 | 1000 | 100 |  625 |
%C A232274 |  108000 |  480 |  510 |  510 | 144 |  289 |
%C A232274 |  217728 |  504 | 1080 | 1440 | 144 |  900 |
%C A232274 |  403368 |  686 | 1470 | 1960 | 196 | 1225 |
%C A232274 |  688128 |  896 | 1920 | 2560 | 256 | 1600 |
%C A232274 | 1102248 | 1134 | 2430 | 3240 | 324 | 2025 |
%C A232274 | 1680000 | 1400 | 3000 | 4000 | 400 | 2500 |
%C A232274 | 1728000 | 1920 | 2040 | 2040 | 576 | 1156 |
%C A232274 +---------+------+------+------+-----+------+
%H A232274 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 A232274 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Circumradius.html">MathWorld: Circumradius</a>
%H A232274 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Inradius.html">MathWorld: Inradius</a>
%e A232274 a(1) = 168 because, for (a,b,c) = (14, 30, 40) => s= (14 + 30 + 40)/2 = 42, and
%e A232274 A = sqrt(42*(42-14)*(42-30)*(42-40)) = sqrt(28224) = 168;
%e A232274 R = abc/4A = 14*30*40/(4*168) = 25;
%e A232274 r = A/s = 168/42 = 4.
%t A232274 nn=2000;Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a)(s-b)(s-c);If[0<area2&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[Sqrt[area2]/s]]&&IntegerQ[Sqrt[a*b*c/(4*Sqrt[area2])]],Print[Sqrt[area2]," ",a " ",b," ",c," ", Sqrt[area2]/s," ", a*b*c/(4*Sqrt[area2])]]],{a,nn},{b,a},{c,b}]
%Y A232274 Cf. A188158, A208984.
%K A232274 nonn,hard
%O A232274 1,1
%A A232274 _Michel Lagneau_, Nov 22 2013