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

24, 96, 120, 168, 216, 240, 336, 384, 432, 480, 600, 624, 672, 720, 768, 840, 864, 960, 1080, 1176, 1320, 1344, 1512, 1536, 1560, 1680, 1728, 1848, 1920, 1944, 2016, 2040, 2160, 2184, 2304, 2376, 2400, 2496, 2520, 2688, 2856, 2880, 2904, 3000, 3024, 3072, 3240

Offset: 1

Author

Michel Lagneau, Mar 04 2012

Keywords

• nonn

Comments

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, ...}.

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.

Examples

			a(1) = 24 because, for (a,b,c) = (6, 8, 10) => s = (6 + 8 + 10)/2 = 12, and
A = sqrt(12(12-6)(12-8)(12-10)) = sqrt(576) = 24;
R = abc/4A = 480/4*24 = 5;
r = A/p = 24/12 = 2.
		

Links

• Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
• Eric W. Weisstein, MathWorld: Circumradius
• Eric W. Weisstein, MathWorld: Inradius

Crossrefs

Cf. A120062, A188158.

Programs

Maple

with(numtheory):T:=array(1..1000):k:=0:nn:=250: for a from 1
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):

Mathematica

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]