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We obtain  the location of the weighted Fermat–Torricelli point  A  0  for a geodesic triangle △  A  1  A  2  A  3  on the  K -plane (two dimensional sphere  S   K   2  and two dimensional hyperbolic plane  H   K   2 ) by deriving a rational algebraic equation which depends on the variable  z  ≡  sin  ∠  A  0  A  1  A  3 . Thus, by squaring both parts of this rational equation and by repeating this process until we eliminate the radicals with respect to  z , we could derive a polynomial equation  P ( z ) = 0, which yields a discrete (finite) set of solutions with respect to the variable  z . The uniqueness of the weighted Fermat–Torricelli point leads to the unique optimal solution (global minimum point) which corresponds to the finite set of solutions of the roots of the polynomial  P ( z ).

Location of the weighted Fermat–Torricelli point on the K -plane (Part II)