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A228383 Area A of the triangle such that A, the sides, and the inradius are integers. 4

%I #11 Apr 09 2017 03:34:01

%S 6,24,30,36,42,48,54,60,66,84,96,108,114,120,126,132,144,150,156,168,

%T 180,192,198,210,216,240,252,264,270,294,300,324,330,336,360,378,384,

%U 390,396,408,420,432,456,462,468,480,486,504,510,522,528,540,546,570

%N Area A of the triangle such that A, the sides, and the inradius are integers.

%C The sequences A208984 and A185210 are subsequences of this sequence. The corresponding inradius r are 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 4, 3, ...

%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. The inradius r is given by r = A/s.

%C a(n) is divisible by 6 and the squares of the form 36k^2 are in the sequence.

%H 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), pp. 32-34.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Inradius.html">MathWorld: Inradius</a>

%e 24 is in the sequence because for (a, b, c) = (6, 8, 10) => s =(6 + 8 + 10)/2 = 12; A = sqrt(12*(12-6)*(12-8)*(12-10)) = sqrt(576)= 24; r = A/s = 2.

%t 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[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

%Y Cf. A188158, A120572, A210250, A208984, A185210.

%K nonn

%O 1,1

%A _Michel Lagneau_, Aug 21 2013

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