Relationship between two centers of circles in a Venn diagram Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$. 
Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively.
For the given values $n(A)=a$, $n(B)=b$, and $n(A \cap B)=c$, find the relationship of their centers in terms of $a$, $b$, and $c$.
The objective is to draw both inner circles.
 A: I haven't gone into calculations but tried to device an algorithm for the prob


*

*Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$

*From the area of the $2$ inner circle find their respective radi

*Now assign variable center to the second inner circle $B$ let us call it point $X$

*From the radius and center find the equation of each of the inner circles respectively

*From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)

*Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$

*Now find the angle $YOZ$ and $YXZ$ and called it $\alpha$ and $\beta$

*With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$

*now the center of the circle $B$ will depend on the equation
 $S_a + S_b - c = S_{\triangle{YOZ}} + S_{\triangle{YXZ}}$


Now for the relation with outer circle: 


*

*distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.

*$l = R_a+R_b+m$

*$b = \max\{R_a,R_b\}$

*Now calculate the area  $l \cdot b$

*put the condition that area less than area of outer circle S.


this algorithm for relation wiht outer circle can be improved!
A: By the areas, the radius of the circles are $r_a=\sqrt{\frac{a}{\pi}},r_b=\sqrt{\frac{b}{\pi}}$ and $r_c=\sqrt{\frac{c}{\pi}}$.
First case: $c=0$, then such a diagram can be drawn if and only if $r_a+r_b\leq r_c$. Pick two centers on the diameter of $S$ which are $r_a$ and $r_b$ far away from the boundary, for example.
Seconda case: $c>0$. 
Let $r$ be the distance between the center. The angle $\phi$ and $\psi$ as below satisfy the equations: (i) $r_a\sin\phi=r_b\sin\psi$ (ii) $c=\phi\frac{a}{\pi}+\psi\frac{b}{\pi}-rr_a\sin\phi$ (iii) $r=r_a\cos\phi+r_b\cos\psi$. Knowing $a,b$ and $c$, we can solve for $\phi,\psi$ and $r$ (perhaps numerically).
Once $r$ is known, such a diagram can be drawn if and only if $r_a+r_b+r\leq 2r_c$ (pick one center $r_a$ far away from the boundary, and the other center $r$ far away from it, the two on a diameter of $S$).

