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In this quadrangle, if $x, y, a, b$ is determined,

$c$ has to be determined too.

Can $c$ be expressed using $x, y, a, b$?

Searching similar questions in axiomatic geometry is hard.

If similar question is present, please let me know.

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  • $\begingroup$ What have you tried? Hint: there are several triangles whose angles are all known. $\endgroup$
    – dxiv
    Aug 22, 2021 at 1:59
  • $\begingroup$ @dxiv I have tried making a triangle EBC (E is a new dot that makes EBC a triangle, which is at the upper side of ABCD) Can you be more specific about the hint? $\endgroup$
    – SGKw
    Aug 22, 2021 at 7:00
  • $\begingroup$ Let $\,BC=d\,$, then you can calculate all sides and diagonals in terms of $\,d\,$ and the given angles, then you can solve one of the triangles for the $\,c\,$ angle. $\endgroup$
    – dxiv
    Aug 22, 2021 at 7:17
  • $\begingroup$ @dxiv I assume letting $BC=d$ is not pure axiomatic geometry...? If you could hint or answer with drawings, that would be grateful. Thank you. $\endgroup$
    – SGKw
    Aug 23, 2021 at 0:13
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    $\begingroup$ @dxiv Thank you for your explanation, now I know that only in special circumstances, $c $ can be calculated without using the law of sines/cosines and trig functions. I posted a new question related to this math.stackexchange.com/questions/4230763/… $\endgroup$
    – SGKw
    Aug 23, 2021 at 2:36

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