 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Cyclic Hexagon And Ratio (Posted on 2007-12-08) 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.) Solution if |UP| = |QR| Comment 6 of 6 | `Let w measure the inscribed angles subtended bysides PQ, RS, and TU. Let x, y , and z measurethe 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 Please log in:
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