
  You want to solve this nonlinear system of equations :
       (with A, B, C  [0..pi/2[ ]pi/2..3pi/2[ ]3pi/2..2pi])
 
        a tan(A) + b tan(B) + c tan(C) = d
        e tan(A) + f tan(B) + g tan(C) = h
        i tan(A) + j tan(B) + k tan(C) = l

  If you write :

               tan(A) = x
               tan(B) = y
               tan(C) = z
    
  You can rewrite the system :

        a x + b y + c z = d
        e x + f y + g z = h
        i x + j y + k z = l

 Which is a linear system of equation in x, y, z.

 Suppose, the solution of the system is :

                    x = r
                    y = s
                    z = t 

 you can write
               tan(A) = r    if r = 0    A = 0 or A = pi or A = 2pi  
               tan(B) = s    ...
               tan(C) = t    ...
 
 The good values for r, s, t are : -1, 0, 1.

 The other values are difficult to write with fractions.