$$ P(x,y,z)\ A(x_1,y_1,z_1)\ B(x_2,y_2,z_2)\ C(x_3,y_3,z_3)\
\vec{OP} = m \cdot \vec{AB} + n \cdot \vec{AC} +\vec{OA}\
\begin{cases} x = m(x_2-x_1)+n(x_3-x_1) + x_1 \ y = m(y_2-y_1)+n(y_3-y_1) + y_1 \ z = m(z_2-z_1)+n(z_3-z_1) + z_1 \ \end{cases} \to \begin{cases} x = mx_2+nx_3 + x_1(1-m-n) \ y = my_2+ny_3 + y_1(1-m-n) \ z = mz_2+nz_3 + z_1(1-m-n) \ \end{cases}
$$