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?

Yes.

link

If you choose as an angle between

aanddyou choose as an angle betweenbandc, 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

aandd, then you know one diagonal, so then you can find an angle betweenbandc.If you choose as an angel between

aanddthen the angle betweenbandcis .