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Locus of Points (Posted on 2007-12-12) Difficulty: 4 of 5
Let X1, X2, ... , Xn 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

   ∑ni=1 |XiP|/|PYi| = n

,where the line XiP also intersects C at point Yi.

See The Solution Submitted by Bractals    
Rating: 3.0000 (2 votes)

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three points again | Comment 6 of 16 |

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.


                                           ------- -------                                          
                                      ---                   ---                                     
                                   -                             -                                  
                                -                                   -                               
                              -                                       -                             
                            -                                           -                           
                          -                                               -                         
                        -                                                   -                       
                                                                                                    
                     -                                                         -                    
                        ********                                     ********                       
                     ***************                             ***************                    
                 - ********************                       ******************** -                
                -*************************                 *************************-               
               -******************************         ******************************-              
              -***********************************************************************-             
             -*************************************************************************-            
            -***************************************************************************-           
             ***************************************************************************            
            *****************************************************************************           
            *****************************************************************************           
         - ******************************************************************************* -        
           *******************************************************************************          
           *******************************************************************************          
       -  *********************************************************************************  -      
          *********************************************************************************         
      -   *********************************************************************************   -     
          *********************************************************************************         
     -    *********************************************************************************    -    
          *********************************************************************************         
    -     *********************************************************************************     -   
           *******************************************************************************          
   -       *******************************************************************************       -  
           *******************************************************************************          
            *****************************************************************************           
  -         *****************************************************************************         - 
            *****************************************************************************           
             ***************************************************************************            
 -            *************************************************************************            -
 -            *************************************************************************            -
 -             ***********************************************************************             -
                *********************************************************************               
                 *******************************************************************                
-                 *****************************************************************                 -
-                   *************************************************************                   -
-                    ***********************************************************                    -
-                      **************************...**************************                      -
-                        **********************..***..**********************                        -
-                           *******************.*****.*******************                           -
-                               **************.*******.**************                               -
-                                             . ***** .                                             -
-                                             .       .                                             -
-                                              .     .                                              -
-                                              ..   ..                                              -
-                                                ...                                                -
-                                                                                                   -
-                                                                                                   -
-                                                                                                   -
                                                                                                    
                                                                                                    
 -                                                                                                 -
 -                                                                                                 -
 -                                                                                                 -
                                                                                                    
                                                                                                    
  -                                                                                               - 
                                                                                                    
                                                                                                    
   -                                                                                             -  
                                                                                                    
    -                                                                                           -   
                                                                                                    
     -                                                                                         -    
                                                                                                    
      -                                                                                       -     
                                                                                                    
       -                                                                                     -      
                                                                                                    
                                                                                                    
         -                                                                                 -        
                                                                                                    
                                                                                                    
                                                                                                    
            -                                                                           -           
             -                                                                         -            
              -                                                                       -             
               -                                                                     -              
                -                                                                   -               
                 -                                                                 -                
                                                                                                    
                                                                                                    
                     -                                                         -                    
                                                                                                    
                        -                                                   -                       
                          -                                               -                         
                            -                                           -                           
                              -                                       -                             
                                -                                   -                               
                                   -                             -                                  
                                      ---                   ---                                     
                                           ------- -------                                          
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                                                                                    
                                           ------- -------                                          
                                      ---                   ---                                     
                                   -                             -                                  
                                -                                   -                               
                              -                                       -                             
                            -                                           -                           
                          -                                               -                         
                        -                                                   -                       
                                                                                                    
                     -                                                         -                    
                                                                                                    
                       ***                                                 ***                      
                 - ************                                        *********** -                
                -****************                                   ****************-               
               -*******************                               *******************-              
              -*********************                             *********************-             
             -************************                         ************************-            
            -**************************                       **************************-           
             ***************************                     ***************************            
            ******************************                  *****************************           
            *******************************               *******************************           
         -  ********************************             ********************************  -        
            **********************************         **********************************           
           **************************************   **************************************          
       -   *******************************************************************************   -      
           *******************************************************************************          
      -     *****************************************************************************     -     
            *****************************************************************************           
     -      *****************************************************************************      -    
            *****************************************************************************     &nbs
  Posted by Charlie on 2007-12-13 00:18:04
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