# Given three circles with non-empty two-way intersections, what is the condition that their three-way intersection is empty?

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.

• 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. – HenrikRüping May 14 at 17:58