OFFSET
1,1
COMMENTS
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).
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.
The product r*R is given by r*R = abc/2(a+b+c).
The following table gives the first values (A, a, b, c, r, R, r*R).
-----------------------------------------------------
| A | a | b | c | r | R | r*R |
-----------------------------------------------------
| 42 | 7 | 15 | 20 | 2 | 25/2 | 5^2 |
| 168 | 14 | 30 | 40 | 4 | 25 | 10^2 |
| 378 | 21 | 45 | 60 | 6 | 75/2 | 15^2 |
| 672 | 28 | 60 | 80 | 8 | 50 | 20^2 |
| 1050 | 35 | 75 | 100 | 10 | 125/2 | 25^2 |
| 1512 | 42 | 90 | 120 | 12 | 75 | 30^2 |
| 2058 | 49 | 105 | 140 | 14 | 175/2 | 35^2 |
| 2688 | 56 | 120 | 160 | 16 | 100 | 40^2 |
| 3000 | 80 | 85 | 85 | 24 | 289/6 | 34^2 |
| 3402 | 63 | 135 | 180 | 18 | 225/2 | 45^2 |
| 4200 | 70 | 150 | 200 | 20 | 125 | 50^2 |
| 5082 | 77 | 165 | 220 | 22 | 275/2 | 55^2 |
| 6048 | 84 | 180 | 240 | 24 | 150 | 60^2 |
| 6960 | 58 | 300 | 338 | 20 | 845/4 | 65^2 |
| 7098 | 91 | 195 | 260 | 26 | 325/2 | 65^2 |
....................................................
REFERENCES
Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.
LINKS
Zak Seidov, Table of n, a(n) for n = 1..100
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric W. Weisstein, MathWorld: Circumradius
Eric W. Weisstein, MathWorld: Inradius
EXAMPLE
a(1) = 42 because, for (a,b,c) = (7, 15, 20):
the semiperimeter s = (7+15+20)/2 =21, and
A = sqrt(21*(21-7)*(21-15)*(21-20)) = 42
R = abc/4A = 7*15*20/(4*42) = 25/2
r = A/s = 42/21 = 2, hence r*R = 25 is a square.
MATHEMATICA
nn=800; lst={}; Do[s=(a+b+c)/2; rr=a*b*c/(2*(a+b+c))
; 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]
PROG
(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
(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),
if((a*b*c)%(2*(a+b+c))<1&&if(issquare(a*b*c/(2*(a+b+c)), &d),
print([A, a, b, c, s, d]))))))))} \\ Faster version uesd for afile. Zak Seidov, Jun 06 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Nov 22 2013
EXTENSIONS
Missing term 33408 added by Zak Seidov, Jun 08 2015
STATUS
approved