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A232329 Integer areas A of the integer-sided triangles such that the product of the inradius and the circumradius is a square.

%I A232329 #43 Sep 05 2025 02:01:32
%S A232329 42,168,378,672,1050,1512,2058,2088,2688,3000,3402,4200,5082,6048,
%T A232329 6960,7098,8232,8352,9450,10752,12000,12138,13608,15162,16800,18522,
%U A232329 18792,20328,22218,24192,26250,27000,27840,28392,30618,31416,32928,33408,35322,36000,37800,40362
%N A232329 Integer areas A of the integer-sided triangles such that the product of the inradius and the circumradius is a square.
%C A232329 The areas of the primitive triangles of sides (a, b, c) and inradius, circumradius equals respectively to r and R are 42, 3000,...  The sides of the nonprimitive triangles are of the form (a*k, b*k, c*k) with r’ = r*k and R’=R*k 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^2. The set {A016850} (numbers (5n)^2) is included in the set of the products r*R (see the table below).
%C A232329 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 A232329 The product r*R is given by r*R = abc/2(a+b+c).
%C A232329 The following table gives the first values (A, a, b, c, r, R, r*R).
%C A232329   -----------------------------------------------------
%C A232329   |    A  |   a  |   b |   c   |   r  |   R    | r*R  |
%C A232329   -----------------------------------------------------
%C A232329   |    42 |   7  |  15  |  20  |    2 |  25/2  |  5^2 |
%C A232329   |   168 |  14  |  30  |  40  |    4 |  25    | 10^2 |
%C A232329   |   378 |  21  |  45  |  60  |    6 |  75/2  | 15^2 |
%C A232329   |   672 |  28  |  60  |  80  |    8 |  50    | 20^2 |
%C A232329   |  1050 |  35  |  75  | 100  |   10 | 125/2  | 25^2 |
%C A232329   |  1512 |  42  |  90  | 120  |   12 |  75    | 30^2 |
%C A232329   |  2058 |  49  | 105  | 140  |   14 | 175/2  | 35^2 |
%C A232329   |  2688 |  56  | 120  | 160  |   16 | 100    | 40^2 |
%C A232329   |  3000 |  80  |  85  |  85  |   24 | 289/6  | 34^2 |
%C A232329   |  3402 |  63  | 135  | 180  |   18 | 225/2  | 45^2 |
%C A232329   |  4200 |  70  | 150  | 200  |   20 | 125    | 50^2 |
%C A232329   |  5082 |  77  | 165  | 220  |   22 | 275/2  | 55^2 |
%C A232329   |  6048 |  84  | 180  | 240  |   24 | 150    | 60^2 |
%C A232329   |  6960 |  58  | 300  | 338  |   20 | 845/4  | 65^2 |
%C A232329   |  7098 |  91  | 195  | 260  |   26 | 325/2  | 65^2 |
%C A232329   ....................................................
%D A232329 Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.
%H A232329 Zak Seidov, <a href="/A232329/b232329.txt">Table of n, a(n) for n = 1..100</a>
%H A232329 Zak Seidov, <a href="/A232329/a232329.txt">Table of 814 values of area A, sides a>=b>=c, semiprime s, and sqrt(Rr), in the order of increasing "a" from 20 up to 10000.</a>
%H A232329 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 A232329 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Circumradius.html">Circumradius</a>
%H A232329 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Inradius.html">Inradius</a>
%e A232329 a(1) = 42 because, for (a,b,c) = (7, 15, 20):
%e A232329   the semiperimeter s = (7+15+20)/2 =21, and
%e A232329   A = sqrt(21*(21-7)*(21-15)*(21-20)) = 42
%e A232329   R = abc/4A = 7*15*20/(4*42) = 25/2
%e A232329   r = A/s = 42/21 = 2, hence r*R = 25 is a square.
%t A232329 nn=800;lst={};Do[s=(a+b+c)/2;rr=a*b*c/(2*(a+b+c))
%t A232329 ;If[IntegerQ[s],area2=s(s-a)(s-b)(s-c);If[0<area2&&IntegerQ[Sqrt[area2]] &&IntegerQ[Sqrt[rr]],AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]
%o A232329 (PARI) lista(nn)=lst=[]; for (a = 1, nn, for (b=1, a, for (c=1, b, s=(a+b+c)/2; rr=a*b*c/(2*(a+b+c)); if ((type(s) == "t_INT") && (type(rr) == "t_INT"), area2=s*(s-a)*(s-b)*(s-c); if ((0<area2) && issquare(area2) && issquare(rr), lst = concat(lst, sqrtint(area2));););););); Set(lst); \\ after Mathematica; _Michel Marcus_, Jun 09 2015
%o A232329 (PARI) {for(a=20,10000,forstep(b=a,2,-1,forstep(c=min(b,a+b-1),a-b+1,-1,if((a+b+c)%2<1,s=(a+b+c)/2;if(issquare(s*(s-a)*(s-b)*(s-c),&A),
%o A232329 if((a*b*c)%(2*(a+b+c))<1&&if(issquare(a*b*c/(2*(a+b+c)),&d),
%o A232329 print([A,a,b,c,s,d]))))))))} \\ Faster version used for afile. _Zak Seidov_, Jun 06 2015
%Y A232329 Cf. A016850, A188158, A208984, A232274.
%K A232329 nonn,changed
%O A232329 1,1
%A A232329 _Michel Lagneau_, Nov 22 2013
%E A232329 Missing term 33408 added by _Zak Seidov_, Jun 08 2015