[Math] Relationship between two centers of circles in a Venn diagram

Question

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.

Answer 1

I haven't gone into calculations but tried to device an algorithm for the prob

1. Fix an origin at one of the inner circle's center say of $A$ and let us call it point $O$
2. From the area of the $2$ inner circle find their respective radi
3. Now assign variable center to the second inner circle $B$ let us call it point $X$
4. From the radius and center find the equation of each of the inner circles respectively
5. From the equation of the $2$ inner circles find the equation of 'chord of contact'(also apply the intersection constraint for the same)
6. Find the points of intersection of the the chord of the contact and any of the $2$ circles and call it points $Y$ and $Z$
7. Now find the angle $YOZ$ and $YXZ$ and called it $\alpha$ and $\beta$
8. With the help of the angles above find area of sector $YOZ$ and $YXZ$ and call it $S_a$ and $S_b$
9. 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:

1. distance between the two centers of the inner circles and call it as $m$. Let their radii are $R_a$, $R_b$.
2. $l = R_a+R_b+m$
3. $b = \max\{R_a,R_b\}$
4. Now calculate the area $l \cdot b$
5. put the condition that area less than area of outer circle S.

this algorithm for relation wiht outer circle can be improved!