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 Locus of Points (Posted on 2007-12-12)
Let X1, X2, ... , Xn be n≥2 distinct points on a circle C with center O and radius r. What is the locus of points P inside C such that

∑ni=1 |XiP|/|PYi| = n

,where the line XiP also intersects C at point Yi.

 See The Solution Submitted by Bractals Rating: 3.0000 (2 votes)

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 three regularly spaced points | Comment 4 of 16 |

With three equally spaced points there are threee lobes.  I don't know if these correspond to polynomial or Erdos lemniscates (http://en.wikipedia.org/wiki/Polynomial_lemniscate), but that idea is intriguing.

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*****************                                             ***************** -                      *************                                                 *************                            ********                                                       ********                                                                                                                                -                                                                           -                         -                                                                         -                           -                                                                       -                             -                                                                     -                               -                                                                   -                                 -                                                                 -                                                                                                                                                                                                                                                -                                                         -                                                                                                                                                  -                                                   -                                                  -                                               -                                                      -                                           -                                                          -                                       -                                                              -                                   -                                                                   -                             -                                                                         ---                   ---                                                                                 ------- -------                                           `

 Posted by Charlie on 2007-12-12 22:06:32

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