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Given two circles $A$ and $B$ with distance from their radius $d$ , is it possible to calculate the radius of the circles? If yes, how?

two circle with their distance

What if we add a $3^{rd}$ circle $C$ , with distance $d_2$ from center of $B$ and $C$ and given that $C$ is half of $A$ i.e $ r_a = 2r_c$ , is it mathematically possible? 3 circles

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  • $\begingroup$ This is not clear. In the first diagram, are you saying that the distance from $A$ to $B$, presumably the two centers, is $d$? If so, then that is not enough to determine the two radii. All you know is that $r_A+r_B=d$. $\endgroup$ – lulu Aug 21 '20 at 19:42
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    $\begingroup$ In first case, you cannot. You can draw many circles with different radii meeting the distance d condition. In 2nd case, you can find it. $\endgroup$ – Math Lover Aug 21 '20 at 19:50
  • $\begingroup$ thanks @MathLover and lulu , i suspected that the first one was lacking information... $\endgroup$ – Akash Jain Aug 21 '20 at 20:14
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In the first example, the only restriction you have is $r_A + r_B = d$. There are infinitely many possibilities for the two radii so you cannot determine them

In the second example, notice that $d = r_A + r_B = 2r_C + r_B$. Therefore, $d - d_2 = r_C$.

Now, that you know $r_C$, you can compute $r_A = 2r_c$ and $r_B = d_2 - r_C$.

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