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Two dissimilar circles intersect at two points. The intersection points in the smaller circle are along its diameter. We know r1, the smaller circle's radius, and a, the distance between the two circles at their furthest separation. We need to solve for r2, the larger circle's radius, in terms of the other two values.

Intersecting Circles

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  • $\begingroup$ Every circles are similar... $\endgroup$ – zoli Nov 2 '17 at 18:11
  • $\begingroup$ OK, so then what I meant was, two circles of different sizes. $\endgroup$ – Paul Holmes Nov 2 '17 at 18:15
  • $\begingroup$ Thanks. Thanks. $\endgroup$ – zoli Nov 2 '17 at 18:20
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$a + r_1 = r_2 + \sqrt {r_2^2 - r_1^2}\\ a + r_1 - r_2 = \sqrt {r_2^2 - r_1^2}\\ a^2 + r_1^2 + r_2^2 + 2ar_1 - 2ar_2 - 2r_1r_2 = r_2^2 - r_1^2 a^2 + 2r_1^2 + 2ar_1 = 2(a + r_2)r_1\\ r_2 = \frac {a^2 + 2r_1^2 + 2ar_1}{2(a+r_1)} \\ $

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For a constructive approach, connect the two points and construct a perpendicular bisector between them. This produces a line that divides both circles in half. Bisect the appropriate chord and you have the radius.

For a mathematical approach, think about the constructive approach and try applying trigonometry.

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Draw a line from $O_2$ to $P$ (which has length $r_2$) and the use of Pythagoras becomes obvious.

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