The six sides of a cyclic hexagon PQRSTU are PQ, QR, RS, ST, TU and UP. It is known that PQ= RS= TU and the diagonals PS, QT and RU meet at the point V. The lines PS and RT intersect at the point W.

Determine the ratio RW/WT, given that PR = 3*RT.

                         .  . .  .  . .  .
                    .                         .
                 .                               .
              .                                     .
            .                                         .
          .                                             .
        .Q                                              R .
       .                                                   .
      .                                                     .
     .                                                       .
    .                                                         .

   .                                                           .
   .                                                           .
   .                                                    V     WS
   P                                                           T
   .                                                           .

    .                                                         .
     .                                                       .
      .                                                     .
       .                                                   .
        .                                                 .
          .                                             .
            .                                         .U
              .                                     .
                 .                               .
                    .                         .
                         .  . .  .  . .  .

Unfortunately the reader must visualise the lineal connections.

(In reply to More Thoughts by Brian Smith)

Without a Geometer's Sketchpad it is difficult to find where the diagonals PS, QT and RU might meet at point V and where PR = 3*RT, and I suspect it might be easier to algebraically and trigonometrically solve this.  Nonetheless, using Paint as my "Geometer's Sketchpad" I can give what I believe might be an approximate ratio -- 8:1.  Unfortunately I can not yet prove this nor be certain how close to accurate my guess is.

 

Edited on December 11, 2007, 3:53 am

Posted by Dej Mar on 2007-12-10 06:43:44