Equation to get the center point of the union of n ellipses?

Question

If I have 3 ellipses that all intersect such as in image.

How can I get the center point of the Union of all three ellipses? (Basically the center point of the red area in the image)

Answer 1

How about we reverse engineer the problem a bit, what do you say ? :-$)$ Instead of going from the equation of the ellipse$($s$)$ to determining the position of the central point, let us consider the center in question fixed at $(0,0)$, and write the equations of the three ellipses in terms of it. Let one of the two axes of symmetry of the first ellipse be the interval $(-a,b)\in\mathbb R$, with $-a<0<b$. Its center lies at $c=\dfrac{a+b}2$, and its equation is $(x-c)^2+\bigg(\dfrac yr\bigg)^2=c^2$, where $r>0$ is a parameter. Then the other two ellipses are the rotations of the first with an angle of $\pm~\dfrac{2\pi}3$ radians. As a fun exercise, draw the $3\times3=9$ ellipses corresponding to $a=1$, $b=5$, $r\in\bigg\{\dfrac25,~1,~\dfrac74\bigg\}$ using Desmos. :-$)$