Let X₁, X₂, ... , X_n 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

   ∑ⁿ_i=1 |X_iP|/|PY_i| = n

,where the line X_iP also intersects C at point Y_i.

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