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.)
re: exploration with computer -- 2 non-opposing points | Comment 2 of 16 |
(In reply to exploration with computer by Charlie)

When the two points in the two point case are separated by less than 180 degrees, as in the below diagram with a separation of 90 degrees, the lemniscate transforms into a kidney shaped blob (remember, it's the border that forms the locus, not the interior). Due to the vertical stretching/horizontal compression, the below does not look symmetrical, but is indeed so about the line y=x. Again, the circle of .'s is superimposed to mark where the center of the circle is; actual *'s and blanks underly them.

As usual, the locus goes through the center, O, and is tangent to the circle C at the designated points Xn, except that at points Xn, that one point (the point of tangency) is missing from the actually defined locus as the ratio of lengths is not defined there.


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

  Posted by Charlie on 2007-12-12 19:52:15
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 (3)
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