# How to find the primitive Heron triangles with three rational angle-bisectors by MMA?

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?

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

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

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}]]


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.

• How is this answer related to writing Mathematica code in order to compute the Heron triangles?
– user49048
Commented Mar 12, 2022 at 4:15