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Inversion in the unit sphere, for a vector $x$, is defined by

$$\frac1x = \frac x{x^2} = \Big(\frac1{x\cdot x}\Big)x$$

How can we prove that a circle's inversion in the sphere is also a circle?

(I already have my own proof, but I didn't find this question elsewhere and thought it should exist. Feel free to improve the question or provide a different proof.)

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