2
$\begingroup$

I'd like to find the primitive Heron triangles with three rational angle-bisectors.

In a Chinese book, I saw these primitive Heron triangles with three sides no longer than 1000, but I'm not sure if they are complete.

How to find these primitive Heron triangles by MMA?

![enter image description here

Certainly, I also searched in the OEIS, but I didn't get useful results.

http://mathworld.wolfram.com/HeronianTriangle.html

$\endgroup$
1

2 Answers 2

2
$\begingroup$
AbsoluteTiming[nn = 1000; lst = {};
 Do[s = (a + b + c)/2;
  If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c);
   l1 = b*c*(b + c + a) (b + c - a)/(b + c)^2;
   l2 = a*c*(a + c + b) (a + c - b)/(a + c)^2;
   l3 = a*b*(a + b + c) (a + b - c)/(a + b)^2;
   If[0 < area2 && 
     IntegerQ[
      Sqrt[area2]] && (Element[Sqrt[l1], Rationals] && 
   Element[Sqrt[l2], Rationals] && Element[Sqrt[l3], Rationals]) &&
  GCD[a, b, c] == 1, AppendTo[lst, {a, b, c}]]], {a, nn}, 
     {b, a}, {c, b}];
   Grid[Transpose[
Partition[
Multicolumn[#, 1] & /@ 
 Partition[SortBy[Sort /@ Union[lst], # &], 5, 5, {1, 1}, 
  Style[{}, White]], 2]], Spacings -> {2, 2}]]

enter image description here

$\endgroup$
0
$\begingroup$

See page 24 and following of Dr Ralph Buchholz's doctoral thesis "On Triangles" for bis triangles with rational area. https://www.ralphthetriangle.com/papers/on-triangles Table 2.1 is a small table of these special Heron triangles.

$\endgroup$
2
  • $\begingroup$ How is this answer related to writing Mathematica code in order to compute the Heron triangles? $\endgroup$
    – user49048
    Commented Mar 12, 2022 at 4:15
  • 1
    $\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$
    – Community Bot
    Commented Mar 12, 2022 at 4:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.