All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
three regularly spaced points | Comment 4 of 16 |

With three equally spaced points there are threee lobes.  I don't know if these correspond to polynomial or Erdos lemniscates (http://en.wikipedia.org/wiki/Polynomial_lemniscate), but that idea is intriguing.


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

  Posted by Charlie on 2007-12-12 22:06:32
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (9)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information