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Let ABCD is a tetragon with its cyclic (outer circle). Diagonal BD bisects angle ABC. The intersection point of BD and AC diagonals is point E. BC = 20, CD = 15, CE=12.

Please help me find AD, ED, angle BCD and the area of tetragon ABCD.

I have tried to find the relationship between the half angles of B, with side AD, but in fact I could not find any.

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  • $\begingroup$ What have you done so far? Showing what you have done and where you are stuck better allows us to tailor an answer to your background and situation. It also demonstrates that you aren't just looking for someone to do your homework for you (that isn't the point of StackExchange). $\endgroup$ – Ian Miller Mar 20 '16 at 15:08
  • $\begingroup$ I have tried to find the relationship between the half angles of B, with side AD, but in fact I could not find any. Moreover this is not my homework, I just like solving math problems. $\endgroup$ – Marie Mar 20 '16 at 15:30
  • $\begingroup$ "with its outer circle" means its cyclic? $\endgroup$ – N.S.JOHN Mar 20 '16 at 15:31
  • $\begingroup$ Yes sure, I mean cyclic. $\endgroup$ – Marie Mar 20 '16 at 15:32
  • $\begingroup$ What does "BD halve angle B" mean? Does it mean that the diagonal BD bisects the angle ABC? $\endgroup$ – coffeemath Mar 20 '16 at 15:40
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Start of an answer: Note that since equal angles subtend equal arcs of the circle, the (shorter) arc CD equals the (shorter) arc DA, since they are subtended respectively by angles CBD and ABD, each half of angle ABC. And the chords formed by equal circular arcs are equal, so that chord CD must equal chord AD in length. Since CD=15 that means that also AD=15.

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  • $\begingroup$ Thanks a lot, it really helped me solve the whole problem. $\endgroup$ – Marie Mar 20 '16 at 17:31
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    $\begingroup$ I'd love to see the solution Marie. Any chance you can post it as solution to your question. $\endgroup$ – Ian Miller Mar 20 '16 at 18:05

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