Three uniform circles are externally tangent to one another and internally tangent to the parabola $y = x^{2}$ . Find the magnitude of the radius.

Provide the answer to 5 decimal places.

The answer is 0.64334.

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If the radius vector to either of the points of tangency of the bottom circle with the parabola makes angle $\theta$ with the downward vertical, it follows that $2r\sin\theta \; = \; \tan\theta$ If the radius vector to the point of tangency of either of the top two circles makes an angle $\phi$ with the downward vertical, then $2r(1 + \sin\phi) \; = \; \tan\phi$ These two conditions ensure that we have the right gradients at these points of tangency. We also need to ensure that the upper and lower points of tangency are the correct vertical height apart, and so $r^2\big[(1+\sin\phi)^2 - \sin^2\theta\big] \; = \; r\big(\sqrt{3} + \cos\theta - \cos\phi\big)$ so that $r(1+\sin \phi + \sin\theta)(1 + \sin\phi - \sin\theta) \; = \; \sqrt{3} + \cos\theta - \cos\phi$ Solving these three conditions numerically, we deduce that $r = 0.643337...$ , making the answer $\boxed{0.64334}$ .