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.

(In reply to exploration with computer by Charlie)

When the two points in the two point case are separated by less than 180 degrees, as in the below diagram with a separation of 90 degrees, the lemniscate transforms into a kidney shaped blob (remember, it's the border that forms the locus, not the interior). Due to the vertical stretching/horizontal compression, the below does not look symmetrical, but is indeed so about the line y=x. Again, the circle of .'s is superimposed to mark where the center of the circle is; actual *'s and blanks underly them.

As usual, the locus goes through the center, O, and is tangent to the circle C at the designated points Xn, except that at points Xn, that one point (the point of tangency) is missing from the actually defined locus as the ratio of lengths is not defined there.

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Posted by Charlie on 2007-12-12 19:52:15