With Eq. 36 from the book by Todhunter we can form a system of equations to guarantee area-isometry of the three outer child triangles.

A1  =  (Subscript[C, c]) +  (Subscript[β, 1]) γ/(Subscript[C, s])

A2 =  (Subscript[B, c]) + (Subscript[β, 2]) γ / (Subscript[B, s])

A3 =  
      (Subscript[A, c])  +  ((-b (Subscript[β, 1]) -  1) (-c (Subscript[β, 2])  - 
                  1)) / ((Subscript[A, s]) (b  -  (Subscript[β, 1])) (c  -  (Subscript[β, 2])))

We first require that always two of the outer child triangles are equal.

eq1 = A1 == A2;

eq2 = A1 == A3;

The two equations can now be solved for the β_1 and β_2.

s1  =  Solve[eq1,  (Subscript[β, 1])]

s2  =  Solve[eq2,  (Subscript[β, 2])]

Using s1 in s2 yields

s3 = s2 /. s1

Solving for β_2 gives now the solution. Note that the cubic equation has three solutions but only the first one is always real.

temp   =  (Subscript[β, 2])  /.  s3;

sol  =  
      Solve[(Subscript[β, 2 ]) ==  temp[[1]][[1]] ,  (Subscript[β, 2])]