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Cyclic Hexagon And Ratio (Posted on 2007-12-08) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips Solution if |UP| = |QR| Comment 6 of 6 |

Let w measure the inscribed angles subtended by
sides PQ, RS, and TU. Let x, y , and z measure
the inscribed angles subtended by sides UP, QR,
and ST respectively.
From triangle RVW,
     |RW|         |RW|         |VW|         |VW|
  ---------- = ---------- = ---------- = ----------
   sin(w+x)     sin(RVW)     sin(VRW)      sin(w)
From triangle WVT,
     |WT|         |WT|         |VW|         |VW|
  ---------- = ---------- = ---------- = ----------
   sin(w+z)     sin(WVT)     sin(VTW)      sin(y)
Combining these two gives
   |RW|     sin(y)     sin(w+x)
  ------ = -------- x ----------                 (1)
   |WT|     sin(w)     sin(w+z)
From triangle PRT,
     3|RT|        |PR|         |RT|         |RT|
  ---------- = ---------- = ---------- = ----------
   sin(w+y)     sin(PTR)     sin(RPT)     sin(w+z)
             or
 
   sin(w+y)
  ---------- = 3                                 (2)   
   sin(w+z)
From triangle PRV,
      3|RT|          |PR|         |RV|        |RV| 
  ------------- = ---------- = ---------- = --------
   sin(2w+y+z)     sin(PVR)     sin(RPV)     sin(w)
From triangle RTV,
       |RT|          |RT|         |RV|        |RV|
  ------------- = ---------- = ---------- = --------
   sin(2w+x+z)     sin(RVT)     sin(RTV)     sin(y)
Combining these two gives
  
   sin(y)     sin(2w+y+z)
  -------- x ------------- = 3                   (3)
   sin(w)     sin(2w+x+z)
If |UP| = |QR|, then x = y and
equation (2) becomes
   sin(w+x) 
  ---------- = 3                                 (4)   
   sin(w+z)
and equation (3) becomes
  
   sin(y)   
  -------- = 3                                   (5)
   sin(w)
Plugging (4) and (5) into (1) gives
   |RW|
  ------ = 3*3 = 9 
   |WT|
Now if someone can just prove |UP| = |QR| or x = y.
 

  Posted by Bractals on 2007-12-10 14:01:55
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