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  "body": "\n## Summary\n\nInversion in a circle with center $O$ and radius $r$ sends a point $P \\ne O$ to a point $P'$ on the ray from $O$ through $P$.\nThe point $P'$ is chosen so that $\\abs{OP} \\cdot \\abs{OP'} = r^2$.\nIt swaps the inside and the outside of the circle and fixes the circle pointwise.\nIt maps circles and lines to circles and lines, and it preserves angles.\nThis makes it a basic tool of [[inversive geometry|HFBW3SXQVM]] and of geometric constructions.\n\n## Explanations\n\n### L1 human\n\nInversion is a mirror for circles instead of lines.\nA point close to the center is sent far away, a point far away is sent close to the center, and points on the circle stay put.\nFor the unit circle centered at the origin, the point $(2, 0)$ goes to $(1/2, 0)$, since $2 \\cdot 1/2 = 1$.\nDoing the inversion twice brings every point back, just as reflecting twice in a mirror does.\n",
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