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 Thoughts by Bractals)

I would not have considered the following as a thought from the crude diagram which I originally put forward which is better superceded by this one of Charlie.

I have wondered about the significance of "V".  It looks from Charlie's diagram that "V" is on the major diagonals ... stop!

Major?  This is not a concept that we consider with circles is it?  With grids, yes. 

[By my right of edit I insert - "Why did I not realise, even though tired, diagonals are not diameters! "]

I am not about to look at this further tonight.  I do note that the assumption of PU=QR does not hold up in the diagram, but the diagram is not the answer.

Bedtime!

Additionally, since the given matched points are diagonally opposing, this must give some substance as to why "V" is so located, and therefore why PU=QR in all cases.  Bye.

Edited on December 10, 2007, 9:02 am

Posted by brianjn on 2007-12-09 09:59:45