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[Generalized problem of Apollonius] Е. А. Морозов |
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Аннотация: The problem of Apollonius (3d century BC) is to construct a circle tangent to the three given circles in the plane. Counting the number of solutions is often considered as one of the first questions of enumerative geometry. It turns out that in general position the problem has 8 solutions and, if not all the given circles are tangent at the same point, then this number is maximal possible. This fact has a plenty of proofs using a wide range of methods, from elementary ones to such as Lie sphere geometry and intersection theory. But what happens if one increases the number of given circles? Clearly, counting the number of solutions in general position is not interesting in this case since this number is always zero. However, the question about the maximal possible number of solution still makes sense. It turns out that if not all the given circles are tangent at the same point, then the problem has at most 6 solutions. The proof of this fact leads to beautiful configurations of tangent circles. In the talk I will describe these construction, give precise statements and proofs, and (if time permits) mention other interesting generalizations of the Apollonius' problem. Язык доклада: английский Website: https://us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09 |
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