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.

Adding a third point (X3) off the axis of symmetry results in a disjoint locus, with the smaller piece seeming to pull a section of the larger piece toward it, making the larger piece asymmetrical. Remember again, the locus is the boundary of the *'d area, not the interior. The Xn's are at 0, 80 and 200 degrees, which are then points of tangency except that the actual point of tangency in each case is missing from the locus due to the lack of definition of the ratio of lengths there.

 

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Posted by Charlie on 2007-12-12 22:04:34