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.

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        .Q                                              R .
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   .                                                    V     WS
   P                                                           T
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            .                                         .U
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Unfortunately the reader must visualise the lineal connections.

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