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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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Solution exploration with computer | Comment 1 of 16

The locus will always include point O, the center of C, as each ratio equals 1 there, and so n of them add up to n. The following program calculates the sum for each point on a grid within the circle and plots a color in DOS full-screen graphics. It won't run under Windows Vista, but will run under XP, except for XP 64, apparently, where we're told no DOS programs will run.  The QuickBasic 4.5 IDE is available at http://www.winsite.com/bin/Info?14000000036569, http://www.qbcafe.net/qbc/english/download/compiler/qbasic_compiler.shtml and other sites. Also see the Wikipedia article http://en.wikipedia.org/wiki/QuickBASIC.  When run, you can make r larger (like 150) to see a more detailed view, in color, and in correct proportion.  One character position on the printed output represents one square pixel on the screen so the view looks compressed horizontally and/or stretched vertically. On the screen it shows correct proportions; this was done so as not to increase the comment pane size unreasonably.

DEFDBL A-Z
SCREEN 12
pi = ATN(1) * 4
dr = pi / 180
n = 2
r = 50
ang(1) = 0
ang(2) = 180
FOR i = 1 TO n
 x(i) = r * COS(ang(i) * dr)
 y(i) = r * SIN(ang(i) * dr)
NEXT

CIRCLE (300, 200), r, 7
PSET (300, 200), 12
FOR px = -r + 1 TO r - 1
 ylim = INT(SQR(r * r - px * px))
 FOR py = -ylim TO ylim
  t = 0
  FOR i = 1 TO n
    PSET (300 + px, 200 - py)
    p1x = x(i): p1y = y(i)
'    CIRCLE (300 + p1x, 200 - p1y), 4, 14
    IF p1x = px THEN
      m = 1D+100
    ELSE
      m = (p1y - py) / (p1x - px)
    END IF
    a = m * m + 1
    b = 2 * m * (py - m * px)
    c = m * m * px * px - 2 * m * py * px + py * py - r * r
    IF b * b - 4 * a * c < 0 OR p1x = px THEN
      p2x = (p1x + px) / 2 ' vertical line among three points
      p2y = -p1y
    ELSE
      IF p1x > px THEN
       p2x = (-b - SQR(b * b - 4 * a * c)) / (2 * a)
      ELSE
       p2x = (-b + SQR(b * b - 4 * a * c)) / (2 * a)
      END IF
      p2y = m * (p2x - px) + py
    END IF
    rr = SQR(p2x * p2x + p2y * p2y)
    IF ABS(rr - r) > 10 THEN
'         CIRCLE (300 + p2x, 200 - p2y), 4, 12
         REM  bad point
    END IF
    d2 = SQR((p2y - py) * (p2y - py) + (p2x - px) * (p2x - px))
    d1 = SQR((p1y - py) * (p1y - py) + (p1x - px) * (p1x - px))
    rat = d2 / d1
    t = t + rat
  NEXT i
  SELECT CASE t
    CASE IS < n * .99999
     c = 14
    CASE IS > n * 1.00001
     c = 9
    CASE ELSE
     c = 4
  END SELECT
 'IF m > 1D+299 THEN c = 7
  PSET (300 + px, 200 - py), c
 NEXT py
NEXT px
    CIRCLE (300, 200), 4, 15

FOR th = -pi / 4 TO pi / 4 STEP 1 / 1000
  rho = r * SQR(ABS(COS(2 * th)))
  x = 300 + rho * COS(th)
  y = 200 - rho * SIN(th)
  CIRCLE (x, y), 1, 2
NEXT th
FOR th = 3 * pi / 4 TO 5 * pi / 4 STEP 1 / 1000
  rho = r * SQR(ABS(COS(2 * th)))
  x = 300 + rho * COS(th)
  y = 200 - rho * SIN(th)
  CIRCLE (x, y), 1, 2
NEXT th


OPEN "locuspts.txt" FOR OUTPUT AS #2
FOR row = 200 - 56 TO 200 + 56
FOR col = 300 - 56 TO 300 + 56
  IF POINT(col, row) = 14 OR POINT(col, row) = 0 THEN
   PRINT #2, " ";
  ELSEIF POINT(col, row) = 15 THEN
   PRINT #2, ".";
  ELSEIF POINT(col, row) = 7 THEN
   PRINT #2, "-";
  ELSEIF POINT(col, row) = 2 THEN
   PRINT #2, "X";
  ELSE
   PRINT #2, "*";
  END IF
NEXT
PRINT #2,
NEXT

CLOSE

The above program is set for n = 2 with the two points diametrically opposed, at 0 degrees and 180 degrees.
The below result shows a locus that matches the lemniscate of Bernoulli (the points where the sum of the ratios exceeds n are marked with *; those where the sum is less than 1 are blank, so the locus sought is the boundary between the *'ed area and the blank area)(http://mathworld.wolfram.com/Lemniscate.html or http://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli). The lemniscate is marked by a band of X's, and does lie on the boundary of the *'d area and the blank area. The original circle, C, is marked by hyphens. The center of the circle is marked by a circle of periods (.), which replace any * or blank that may lie under them. As the center is on the locus, it's on the border between the *'s and the blanks, and so the .'s usually replace half *'s and half blanks.
 

                                                                                                      

                                                                                                       
                                            ------- -------                                           
                                       ---                   ---                                      
                                    -                             -                                   
                                 -                                   -                                
                               -                                       -                              
                             -                                           -                            
                           -                                               -                          
                         -                                                   -                        
                                                                                                      
                      -                                                         -                     
                                                                                                      
                                                                                                      
                  -                                                                 -                 
                 -                                                                   -                
                -                                                                     -               
               -                                                                       -              
              -                                                                         -             
             -                                                                           -            
                                                                                                      
                                                                                                      
                                                                                                      
          -                                                                                 -         
                                                                                                      
                                                                                                      
        -                                                                                     -       
                                                                                                      
       -                                                                                       -      
                                                                                                      
      -                                                                                         -     
                                                                                                      
     -                                                                                           -    
                  XXXXXX                                                       XXXXXX                 
    -        XXXXXXXXXXXXXXXX                                             XXXXXXXXXXXXXXXX        -   
           XXXXXXXXXXXXXXXXXXXXX                                       XXXXXXXXXXXXXXXXXXXXX          
         XXXXXXXXXX****XXXXXXXXXXX                                   XXXXXXXXXXX****XXXXXXXXXX        
   -    XXXXXX**************XXXXXXXX                               XXXXXXXX**************XXXXXX    -  
       XXXXX*******************XXXXXX                             XXXXXX*******************XXXXX      
      XXXX***********************XXXXXX                         XXXXXX***********************XXXX     
  -  XXXX**************************XXXXX                       XXXXX**************************XXXX  - 
  - XXXX*****************************XXXXX                   XXXXX*****************************XXXX - 
  -XXXX*******************************XXXXX                 XXXXX*******************************XXXX- 
  XXXX**********************************XXXX               XXXXX*********************************XXXX 
  XXX************************************XXXX             XXXX************************************XXX 
 XXXX*************************************XXXX           XXXX*************************************XXXX
 XXX***************************************XXXX         XXXX***************************************XXX
 XXX****************************************XXXX       X XX****************************************XXX
XXXX*****************************************XXXX ... X X******************************************XXXX
XXX*******************************************XX.X   X.X********************************************XXX
XXX*********************************************X X X X*********************************************XXX
XXX********************************************.*X*X*X*.********************************************XXX
XXX********************************************.**X*X**.********************************************XXX
XXX********************************************.*X*X*X*.********************************************XXX
XXX*********************************************X X X X*********************************************XXX
XXX********************************************X.X   X.XX*******************************************XXX
XXXX******************************************X X ... XXXX*****************************************XXXX
 XXX****************************************XX X       XXXX****************************************XXX
 XXX***************************************XXXX         XXXX***************************************XXX
 XXXX*************************************XXXX           XXXX*************************************XXXX
  XXX************************************XXXX             XXXX************************************XXX 
  XXXX*********************************XXXXX               XXXX**********************************XXXX 
  -XXXX*******************************XXXXX                 XXXXX*******************************XXXX- 
  - XXXX*****************************XXXXX                   XXXXX*****************************XXXX - 
  -  XXXX**************************XXXXX                       XXXXX**************************XXXX  - 
      XXXX***********************XXXXXX                         XXXXXX***********************XXXX     
       XXXXX*******************XXXXXX                             XXXXXX*******************XXXXX      
   -    XXXXXX**************XXXXXXXX                               XXXXXXXX**************XXXXXX    -  
         XXXXXXXXXX****XXXXXXXXXXX                                   XXXXXXXXXXX****XXXXXXXXXX        
           XXXXXXXXXXXXXXXXXXXXX                                       XXXXXXXXXXXXXXXXXXXXX          
    -        XXXXXXXXXXXXXXXX                                             XXXXXXXXXXXXXXXX        -   
                  XXXXXX                                                       XXXXXX                 
     -                                                                                           -    
                                                                                                      
      -                                                                                         -     
                                                                                                      
       -                                                                                       -      
                                                                                                      
        -                                                                                     -       
                                                                                                      
                                                                                                      
          -                                                                                 -         
                                                                                                      
                                                                                                      
                                                                                                      
             -                                                                           -            
              -                                                                         -             
               -                                                                       -              
                -                                                                     -               
                 -                                                                   -                
                  -                                                                 -                 
                                                                                                      
                                                                                                      
                      -                                                         -                     
                                                                                                      
                         -                                                   -                        
                           -                                               -                          
                             -                                           -                            
                               -                                       -                              
                                 -                                   -                                
                                    -                             -                                   
                                       ---                   ---                                      
                                            ------- -------                                            
        

Edited on December 12, 2007, 7:49 pm
  Posted by Charlie on 2007-12-12 19:48:10

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