Given the center coordinates $\{x_i,y_i\}$ and radii ($r_i$) of three circles ($i=1,2,3$) (the areas of which are certain probabilities), the three pairwise intersections being all non-empty, what is the condition that their three-way intersection is, on the other hand, empty?

In particular, I have in mind three circles $A,B,C$ of radii $\frac{8 \pi}{27 \sqrt{3}}$ and $\frac{1}{6}$ and $\frac{1}{6}$. The intersection of $B$ and $C$ is $\frac{1}{9}$ and $A \land B$ and $A \land C$ are both $-\frac{4}{9}+\frac{4 \pi }{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$, while as indicated, $A \land B \land C$ is $\varnothing$.

This problem pertains to my efforts (https://mathematica.stackexchange.com/questions/198019/create-a-venn-diagram-showing-the-relations-of-three-sets-of-quantum-states/198091#198091) to represent--via a Venn or related diagram--the Hilbert-Schmidt probabilities of certain quantum ("positive-partial-transpose" and "bound-entangled") "two-qutrit" states (representable by $9 \times 9$ density matrices).

So, how to locate the three centers--subject to the three circles all being contained in a circle (arbitrarily centered at the origin) of area/probability 1--is also in question (https://math.stackexchange.com/questions/3224845/given-a-circle-a-of-area-1-centered-at-0-0-give-conditions-that-another).

I also have further information as to the probabilities assigned to various unions and intersections of $A,B,C$ and their negations--which might be helpful.

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    $\begingroup$ The intersection is nonempty, iff the midpoint of the incirce of the triangle formed by the three midpoints of the circle lies in this intersection. $\endgroup$ – HenrikRüping May 14 at 17:58

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