OFFSET
1,7
COMMENTS
For n <= 200, the number of distinct integer-sided triangles inscribed in a circle of radius A009003(n) whose inradius are integers belongs to the set E = {1, 5, 10, 12, 38} where a(168) = 38 (see the table given in reference). Is the set E infinite when n is infinite?
We observe geometric properties:
If a(n) = 1, the unique triangle is a right triangle.
If a(n) = 5, we find two right triangles, two isosceles triangles and another triangle (neither isosceles nor right triangle).
If a(n) = 10, we find three right triangles, two isosceles triangles and five other triangles.
If a(n) = 12, we find four right triangles and eight other triangles.
The area A of a triangle whose sides have lengths u, v, and w is given by Heron's formula: A = sqrt(s*(s-u)*(s-v)*(s-w)), where s = (u+v+w)/2.
The inradius r is given by r = A/s and the circumradius is given by R = u*v*w/4A.
LINKS
Felix Huber, Illustration of the term a(7)
Michel Lagneau, Triangles
Eric Weisstein's World of Mathematics, Circumradius
Eric Weisstein's World of Mathematics, Inradius
EXAMPLE
a(7) = 5 because there exists 5 distinct triangles of integer circumradius R = A009003(7)= 25 with the corresponding integer inradius {4, 6, 8, 10, 12}.
MATHEMATICA
A009003=Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &]; lst= {}; Do[R=Part[A009003, n]; it=0; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[area2>0&&IntegerQ[Sqrt[area2]]&&R==a*b*c/(4*Sqrt[area2])&&IntegerQ[Sqrt[area2]/s], it=it+1]], {a, 2*R}, {b, a}, {c, b}]; AppendTo[lst, it], {n, 1, 30}]; lst
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 03 2018
STATUS
approved