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.

Although the puzzle specifies n to be at least 2, when n is 1 there's also a locus that meets the criteria. It's just a circle half the diameter of circle C, with one of its diameters on C and the other at O. The points of the locus are the midpoints of all chords with one end at the given X1. Perhaps this was considered too trivial to be considered to be within the scope of the puzzle.

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Posted by Charlie on 2007-12-13 00:21:20