trying to find 1/2 angle tangent formuli for rationals

[American Citizen <website.reader3@gmail.com>]

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?

Randall