Re: trying to find 1/2 angle tangent formuli for rationals

On Sat 14 Sep 2024 à 01:36, American Citizen <website.reader3@gmail.com> a écrit :

Hello:

It is a known (maybe not so well known) fact that if a person is given
the rational values [a,b] for the 1/2 angle tangents of a triangle, then
the 3rd 1/2 angle tangent is also rational and the triangle is a Heron
triangle.

I am trying to find an algebraic _expression_ in GP Pari, which takes as
input [a,b] (rational or symbolic) and outputs the sides of the triangle.

Example:

for the 13,14,15 Heron triangle, the 1/2 angle tangents are [ 1/2, 4/7,
2/3 ]

So if I input f(1/2,4/7) I should expect to see (intermediate value 2/3
for the missing 1/2 angle tangent) and finally 13,14,15

My goal is to input certain rationals creating square sided Heron
triangles (2 of the sides must be squares) and checking to see if we
have a 3rd square side

I know of only 2 such Heron triangles, 4427,4380,1853 and
68595,68104,11789 and I am searching for a 3rd such example.

I do know that the middle side > 842,180, having exhaustively searched
the 42+ quadrillion triangles.

Can we find an algebraic _expression_ for the triangles sides s,t,u, given
a,b as the 1/2 angle tangents?

This is actually really easy.

First, as I said before, the third tangent is c=(1-ab)/(a+b). Let

S = 2a/(1+a^2), T = 2b/(1+b^2) and U = 2c/(1+c^2) be the

sines of the (full) angles. Then the laws of sines (or whatever you

want to call it) says that [s,t,u] and [S,T,U] are proportional.

f(a,b)=my(c=(1-a*b)/(a+b),v=[2*a/(1+a^2),2*b/(1+b^2),2*c/(1+c^2)]);v/content(v);

Loïc