The question is
"Show that the set of points on the plane determined by the three points $(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)$ is $[(\frac{lx_1+mx_2+nx_3}{l+m+n},\frac{ly_1+my_2+ny_3}{l+m+n},\frac{lz_1+mz_2+nz_3}{l+m+n})]$ such that l+m+n$\ne$0"
Just as a point on a line joining two points $(x_1,y_1,z_1),(x_2,y_2,z_2)$ can be represented as $(\frac{kx_2+x_1}{1+k},\frac{ky_2+y_1}{1+k},\frac{kz_2+z_1}{1+k})$ s.t (k+1)$\ne$0, where k is the parameter . The physical interpretation is that the line joining the two points are divided in the ratio 1:k.
This problem wants to extend this concept to the entire plane with l,m,n as the parameters.The solution must be simple but I am not able to solve it as I am new to analytical geometry