We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$

where $a,b$ are two given points in the plane and $\lambda$ is a constant.

Now we consider the following generalization:

For three given points $a,b,c \in \mathbb{R}^{2}$ define $$A_{\lambda}=\{z\in \mathbb{R}^{2} \;\text{with}\;\; |z-a|+|z-b|+|z-c|=\lambda\}$$

How is the geometric description of $A_{\lambda}$? Is it a closed curve, at least when $\lambda$ is very large?

More generally, assume that $G$ is a compact topological group with Haar measure $\mu$. Assume that $G$ is topologically embedded in the plane. Define:

$$A_{\lambda}=\{z\in \mathbb{R}^{2} \;\text{with}\;\; \int_{G} |z-g|d\mu=\lambda\}$$

Is the topology of $A_{\lambda}$ independent of choosing sufficiently large $\lambda$?