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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

D.Matthew
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2 Answers2

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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

Eufisky
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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.

Randall
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  • How is this answer related to writing Mathematica code in order to compute the Heron triangles? –  Mar 12 '22 at 04:15
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    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. – Community Mar 12 '22 at 04:55