Let $A,B,C$ points that doesn't lie in the same line and which have position vectors $a,b,c$ relative to an origin $O$ respectively. Prove that an equation of the plane which passes through $A,B,C$ is given by:
$$r=\frac{na+mb+pc}{m+n+p}$$
where $m,n,p$ are arbitrary scalars, also prove that this equations is independent of the origin.
I've already proved that $r\cdot[(a-b)\times(b-c)]=0$ but I don't know if this is sufficient and I don't know how to proceed with the independence of the origin, this is a Schaum Vector Analysis Problem. Any help is preciated!
Since we need $s=m+n+p \neq 0$ for $r$ to be defined, we may divide each of $m,n,p$ by $s$ and replace them in the formula for $r,$ which is homogeneous in the triple $m,n,p.$
After this, $m+n+p=1$ and given $a,b,c$ noncollinear we have that the triple $(n,m,p)$ give the barycentric coordinates of $r$ with respect to the base triangle $a,b,c.$