I am told that the set of all $\vec{r}$ for which $|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is true forms a sphere---however, my semi-intuitive reading of this equation puts a "weaker weight" on the second vector which would make me think that this should represent an ellipsoid if anything. I'm not sure of how I would check my thoughts about this, so can anyone confirm what this shape is? If it's a sphere, how would one figure out its center and radius?
If it's an ellipsoid I would assume that $\vec{r}_1,\vec{r}_2$ represent focii.
Let's give a try. For simplicity, let's work in $\Bbb R^2$ and assume WLOG that $\vec r_2=0$. The equation of the figure would be something like: $$2(x-a)^2+2(y-b)^2=x^2+y^2$$ or $$x^2-4ax+2a^2+y^2-4by+2b^2=0$$ which is a circle centered at $(2a,2b)$ and radius $\sqrt{2a^2+2b^2}$.
The general case shouldn't be more difficult.