The question states
$z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the roots of the equation $x^3+3x^2+px+q=0$ then comment on the signs of $p$ and $q$.
Using $l+m+n=-3$, i get $z=-\frac{lz_1+mz_2+nz_3}{3}$.
But I am unable to proceed further.
At first glance, the expression looked similar to the section formula for two points. I'm interested to know if there are any similarities with that.
Also, can conditions be imposed on $l,m,n$ in the general case, that is, without giving $l+m+n=3$?