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.

The following two diagrams illustrate the symmetry of X1 and X2 separated by 90 degrees, and then a distortion when X3 is introduced off-axis. By aligning the axis vertically, the stretching/compression doesn't affect the symmetry/asymmetry.

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                        ********                                     ********                        
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                       ***                                                 ***                       
                 - ************                                        *********** -                 
                -****************                                   ****************-                
               -*******************                               *******************-               
              -*********************                             *********************-              
             -************************                         ************************-             
            -**************************                       **************************-            
             ***************************                     ***************************             
            ******************************                  *****************************            
            *******************************               *******************************            
         -  ********************************             ********************************  -         
            **********************************         **********************************            
           **************************************   **************************************           
       -   *******************************************************************************   -       
           *******************************************************************************           
      -     *****************************************************************************     -      
            *****************************************************************************            
     -      *****************************************************************************      -     
            *****************************************************************************     &nbs		

		

			

			

 			
			

				Posted by Charlie
				on 2007-12-13 00:18:04