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 three points again by Charlie)

                                          ------- -------                                           
                                      ---                   ---                                      
                                   -                             -                                   
                                -                                   -                                
                              -                                       -                              
                            -                                           -                            
                          -                                               -                          
                        -                                                   -                        
                                                                                                     
                     -                                                         -                     
                                                                                                     
                       ***                                                 ***                       
                 - ************                                        *********** -                 
                -****************                                   ****************-                
               -*******************                               *******************-               
              -*********************                             *********************-              
             -************************                         ************************-             
            -**************************                       **************************-            
             ***************************                     ***************************             
            ******************************                  *****************************            
            *******************************               *******************************            
         -  ********************************             ********************************  -         
            **********************************         **********************************            
           **************************************   **************************************           
       -   *******************************************************************************   -       
           *******************************************************************************           
      -     *****************************************************************************     -      
            *****************************************************************************            
     -      *****************************************************************************      -     
            *****************************************************************************            
    -        ***************************************************************************        -    
             ***************************************************************************             
   -          *************************************************************************          -   
              *************************************************************************              
               ***********************************************************************               
  -             *********************************************************************             -  
                 *******************************************************************                 
                  *****************************************************************                  
 -                 ***************************************************************                 - 
 -                  ************************************************************                   - 
 -                   *********************************************************                     - 
                       *****************************************************                         
                        **************************************************                           
-                         ********************************************                              -
-                           ***************************************                                 -
-                            *********************************                                      -
-                              ******************...*******                                         -
-                                **************..***..**                                            -
-                                  ************.*****.                                              -
-                                    *********.*****  .                                             -
-                                      *******.****   .                                             -
-                                        *****.**     .                                             -
-                                          ****.     .                                              -
-                                              ..   ..                                              -
-                                                ...                                                -
-                                                                                                   -
-                                                                                                   -
-                                                                                                   -
                                                                                                     
                                                                                                     
 -                                                                                                 - 
 -                                       *****                                                     - 
 -                                      ********                                                   - 
                                       **********                                                    
                                      ***********                                                    
  -                                  *************                                                -  
                                    ***************                                                  
                                   ****************                                                  
   -                              ******************                                             -   
                                 *******************                                                 
    -                            ********************                                           -    
                                *********************                                                
     -                         ***********************                                         -     
                               ***********************                                               
      -                       ************************                                        -      
                              *************************                                              
       -                     **************************                                      -       
                             **************************                                              
                             **************************                                              
         -                  ***************************                                    -         
                            ***************************                                              
                            ***************************                                              
                            ***************************                                              
            -               ***************************                                 -            
             -              ***************************                                -             
              -             ***************************                               -              
               -            ***************************                              -               
                -           ***************************                             -                
                 -          **************************                             -                 
                            **************************                                               
                             *************************                                               
                     -       ************************                          -                     
                              **********************                                                 
                        -     **********************                        -                        
                          -    ********************                       -                          
                            -   ******************                      -                            
                              -  ****************                     -                              
                                - *************                     -                                
                                   - ********                    -                                   
                                      ---                   ---                                      
                                           ------- -------                                           
                                                                                                     
 

Posted by Charlie on 2007-12-13 00:19:33