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.

  • 2
    $\begingroup$ When you say 'relationship', do you mean distance? Also, apart from giving bounds for $a$, $b$ and $c$, $S$ doesn't really do anything, does it? $\endgroup$ – Karolis Juodelė Jul 12 '12 at 8:26
  • $\begingroup$ You've tried making a drawing? $\endgroup$ – J. M. is a poor mathematician Jul 12 '12 at 8:26
  • $\begingroup$ @J.M.: I haven't drawn it yet because there might be many solutions. $\endgroup$ – kiss my armpit Jul 12 '12 at 8:36
  • $\begingroup$ If you're trying to draw an area-proportional Venn diagram, you could have $a=b=\frac12$ but $c=0$, in which case there is no solution. $\endgroup$ – Rahul Jan 22 '13 at 20:03
  • $\begingroup$ there might be many solutions... Are you sure? For any given values of $a$ and $b$, any $0\lt c\lt\min(a,b)$ uniquely determines the distance between their centers. $\endgroup$ – Did Jan 26 '13 at 14:08

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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.