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  "body": "\n## Problem\n\nLet $\\Gamma_1$, $\\Gamma_2$, $\\Gamma_3$ be three [[circles|060W5CBZP6]] in the Euclidean plane.\nConstruct, with [[straightedge and compass|0DAECMZVRT]], every circle that is tangent to all three of them.\n\n## Remarks\n\nThe problem is solved: straightedge and compass constructions of all solutions are known.\nApollonius posed it in his work *Tangencies*, which is lost, so his own solution is not known in detail.\nFrançois Viète gave a straightedge and compass solution.\nA solution circle may touch each $\\Gamma_i$ from outside or from inside, which gives $2^3 = 8$ cases.\nSo, outside special positions with infinitely many solutions, there are at most eight solution circles, and some positions give fewer or none.\nInversion in a circle and the algebra of three quadratic equations both lead to solutions.\nVariants replace some of the circles by points or by lines.\n\n## Explanations\n\n### L1 human\n\nYou are given three circles drawn on a sheet of paper.\nThe task is to draw a fourth circle that just touches each of the three.\nThere are usually several answers, up to eight, depending on which of the given circles the new circle wraps around.\n",
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