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 Locus of Points (Posted on 2007-12-12)
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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 and also | Comment 8 of 16 |

Although the puzzle specifies n to be at least 2, when n is 1 there's also a locus that meets the criteria. It's just a circle half the diameter of circle C, with one of its diameters on C and the other at O. The points of the locus are the midpoints of all chords with one end at the given X1. Perhaps this was considered too trivial to be considered to be within the scope of the puzzle.

`                                           ------- -------                                                                                 ---                   ---                                                                         -                             -                                                                   -                                   -                                                              -                                       -                                                          -                                           -                                                      -                                               -                                                  -                                                   -                                                                                                                                                  -                                                         -                                                                                                                                                                                                                                                -                                                                 -                                 -                                                                   -                               -                                                                     -                             -                                                                       -                           -                                                                         -                         -                                                                           -                                                                                                                                                                                                                                                                                                                                    -                                                                                 -                                                                                                                                                                                                                          -                                                                                     -                                                                                  *                               -                                                             ***************           -                                                                        *******************                     -                                                          ***********************        -                                                                   ***************************                -                                                       *******************************     -                                                               *********************************            -                                                      ***********************************    -                                                            *************************************                                                               ***************************************        -                                                    *****************************************  -                                                         *****************************************                                                           *******************************************     -                                                    *******************************************  -  -                                                   ********************************************* -  -                                                   ********************************************* -                                                     ***********************************************                                                      ***********************************************  -                                                  *************************************************--                                                  *************************************************--                                                  *************************************************--                                                ...************************************************--                                              ..  *..**********************************************--                                              .   **.**********************************************--                                             .    ***.*********************************************--                                             .   ****.*********************************************--                                             .    ***.*********************************************--                                              .   **.**********************************************--                                              ..  *..**********************************************--                                                ...************************************************--                                                  *************************************************--                                                  *************************************************--                                                  *************************************************-                                                    ***********************************************                                                      ***********************************************   -                                                   ********************************************* -  -                                                   ********************************************* -  -                                                    *******************************************  -                                                       *******************************************                                                           *****************************************       -                                                    *****************************************  -                                                          ***************************************                                                               *************************************          -                                                      ***********************************    -                                                              *********************************             -                                                       *******************************     -                                                                  ***************************                 -                                                          ***********************        -                                                                       *******************                      -                                                             ***************           -                                                                                 *                                -                                                                                     -                                                                                                                                                                                                                          -                                                                                 -                                                                                                                                                                                                                                                                                                                                    -                                                                           -                         -                                                                         -                           -                                                                       -                             -                                                                     -                               -                                                                   -                                 -                                                                 -                                                                                                                                                                                                                                                -                                                         -                                                                                                                                                  -                                                   -                                                  -                                               -                                                      -                                           -                                                          -                                       -                                                              -                                   -                                                                   -                             -                                                                         ---                   ---                                                                                 ------- -------                                           `

 Posted by Charlie on 2007-12-13 00:21:20

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