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 Solution by Praneeth)

As an example, consider the circle x˛+y˛=1, with two given points: X1 = (1,0), and X2 = (-1,0), both lying on the circle.

If the locus sought in this case is a circular arc, what are you claiming is the center and radius of that circular arc?  I'll assume that your claim is that all parts of such a defined circle that lie within the given circle constitute the arc that is the locus; otherwise specify what portion of the newly defined circle is the arc in question.

 

Edited on December 14, 2007, 12:54 pm

Posted by Charlie on 2007-12-14 12:52:55