$|z-1|+|z-2|+|z-3|=6$ in the Argand Plane

Question

In the argand plane $$C:|z-1|+|z-2|+|z-3|=6 ~~~(1)$$ represents a bounded curve which is a rounded blob mimicking an "ellipse".

Using the the inequality $$|Z_1+Z_2+Z_3| \le |Z_1|+|Z_2|+|Z_3|, ~~~~(2)$$ we can see that $$6=|z-1|+|z-2|+|z-3|\ge |3z-6| \implies |z-2| \le 2. ~~~(3)$$ The circle $C_1: |z-2|=2$ touches and remains outside $C$. Next, it will be interesting to use some other inequality or something to find the equation of a bounded curve $C_2$ which touches and remains inside $C$. Eventually, $C$ will touch $C_1$ and $C_2$ and it will be enclosed between them.

The question is to find the curve $C_2.$

Edit:
For $C_2$, the RMS-AM inequality gives: $$\sqrt{\frac{|z-1|^2+|z-2|^2+|z-3|^2}{3}} \ge \frac{|z-1|+|z-2|+|z-3|}{3}=2~~~~(4)$$
The equality in above doesn't holds as $|z-1|=|z-2|=|z-3|$ cannot be met $$\implies(x-2)^2+y^2 >\sqrt{\frac{10}{3}}=1.82574~~~~(5)$$
This circle $|z-2|=\sqrt{\frac{10}{3}}$ cannot be the required $C_2$ as it cannot touch $C$ in (1).

Answer 1

Our claim is that the circle $$C_2:|z-2|=\sqrt\frac{44}{3}-2$$ lies entirely inside $C$. It is not hard to show that this value is the unique real $r$ for which $2\pm ir$ are on $C$.

We can calculate that $$|z-2|^2=\frac{|z-1|^2+|z-3|^2}{2}-1.$$ We have that $$\frac{|z-1|^2+|z-3|^2}{2}\geq\left(\frac{|z-1|+|z-3|}{2}\right)^2,$$ with equality if and only if $|z-1|=|z-3|$, so $$|z-1|+|z-3|\leq 2\sqrt{|z-2|^2+1},$$ again with equality if and only if $|z-1|=|z-3|$. So, for a point $z$ on $C$, $$6=|z-1|+|z-2|+|z-3|\leq |z-2|+2\sqrt{|z-2|^2+1};$$ since the right side is increasing in $|z-2|$, this implies that $|z-2|$ is at least the real $r$ for which $6=r+2\sqrt{r^2+1}$, with equality reached if and only if $z=2\pm ir$. It can be calculated that $r=\sqrt\frac{44}3-2$ is the positive value of $r$ that satisfies $6=r+2\sqrt{r^2+1}$, so we are done.

Answer 2

Let us consider the equivalent problem, with a $(-2,0)$ translation:

$$|z+1|+|z|+|z-1|=6\tag{1}$$

Let (C) be the corresponding curve (generalized elllipse). It is possible to give an intuitive graphical construction of (C) (see Fig. 2) that will naturaly involve the maximal circle. Morover, in this way, we will be able to provide parametric equations of (C) (see (4)).

Due to the invariance of (1) with respect to transformations $z \to -z$ and $z \to $\overline{z}$, it is enough to study (C) in the first quadrant.

Fig. 1: The thick blue curve is the part of (C) belonging to the first quadrant (passing in particular through point $(2,0))$. It has been plotted using parametric equations (4). The other curves are level curves with equations $|z+1|+|z|+|z-1|=k$ for different values of $k$.

Let us introduce a parameter $m$ transforming (1) into the equivalent issue:

$$\begin{cases}\text{Circle }(C_m):&|z|&=&2m\\\text{Ellipse } (E_m):&|z+1|+|z-1|&=&6-2m\end{cases}\tag{2}$$

This is represented on Fig. 2:

Fig. 2: Circles $(C_m)$ and ellipses $(E_m)$ for values of $m \in [0,1]$ from red (small values of) $m$ to blue (values of $m$ close to $1$). Curve (C) in black. In fact $m \in [m_0/2,1]$ where $m_0=\sqrt{\tfrac{44}{3}}-2$... We have found back this critical value! The corresponding circle is thickened : it is the biggest circle that can be included into (C), and tangent to it (a single intersection solution).

Clearly, $m \in [0,3]$. In fact, the domain of variation of $m$ is much more narrow: this is easy to understand on Fig. 2: $m$ cannot be too small, because in this case the circle is itself too small to have an intersection with the ellipse.

The foci of $(E_m)$ are $z=-1$ and $z=1$ ; its semiaxes are:

$$a=3-m \ \ \text{and} \ \ b=\sqrt{a^2-1}.$$

A classical parametrization for the ellipse is:

$$\begin{cases}x&=&a \cos \theta\\y&=&b \sin \theta \end{cases} \ \iff \ \begin{cases}x^2&=&(3-m)^2 \cos^2 \theta\\y^2&=&((3-m)^2-1) \sin^2 \theta \end{cases}\tag{3}$$

If $(x,y)\in(C_m)\cap(E_m)$, we add the two right hand equations:

$$(2m)^2=(3-m)^2-\sin^2 \theta$$

For the first quadrant, where $\sin \theta \ge 0$ and $\cos \theta \ge 0$, we have a unique solution:

$$\sin \theta=\sqrt{(3-m)^2-4m^2}=\sqrt{-3(m-1)(m+3)}\tag{4}$$

meaning that the domain of variation of $m$ is included into $[0,1]$ (in fact, smaller).

We deduce from (3) and (4) the following parametric equations for the locus :

$$\begin{cases}x&=&(3-m)\sqrt{3m^2+6m-8}\\ y&=&\sqrt{-3(4-m)(2-m)(m-1)(m+3)}\end{cases}\tag{5}$$

(valid for the first quadrant as said upwards).

Remark: the critical value of the radius can be obtained (see legend of Fig. 2) by setting $x=0$ in (4), i.e. by solving $3m^2+6m-8=0$.