Can we always form a cyclic quadrilateral using 4 side lengths? I was solving this problem on SPOJ http://www.spoj.com/problems/QUADAREA/. In order to find the maximum area of quadrilateral with side lengths given, I applied Brahmagupta's formula K={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\, and i got AC. I am wondering, is that always possible? Is there any proof?
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Yes.
link
If you choose
as an angle between a and d you choose 
as an angle between b and c, then you have cyclic quadrilateral.
link
using this, you can prove that it will be valid quadrilateral by finding diagonal using a and d and using b and c
But u can control only one angle at a time I guess, either between a and d or between b and c, provided a and d, b and c are adjacent.
Yes, if you control an angle between a and d, then you know one diagonal, so then you can find an angle between b and c.
If you choose
as an angel between a and d then the angle between b and c is 
.