Given three complex number $u, v, z$ consider all complex numbers of the form $au + bv + cz$ with $a, b, c$ being non-negative real numbers satisfying $a + b + c = 1$. What geometric shape is formed by all these points?
I'm not sure how I should even start on this question. Suppose, $a=1,b=0,c=0$ and all its rotations. Then the three points would be $u,v,z$. I think the locus COULD be a triangle with vertices $u,v,z$? Thanks!
Yes, under given conditions it's called a convex combination of points and in general it is indeed a triangle with vertices $u,v,z$, but it also could turn out to be a line segment or even a point (when?).