Brain Wave

So the problem statement is this:
"Prove that the area of a quadrilateral with sides,in cyclic order, are a, b , c ,d. (here a and c are opposite to each other, please note), is less than 1/2(a*c + b*d)"
You might want to try it before reading ahead. This was a problem in an old Balkan national olympiad and i wondered how the hell they gave such a simple problem in a national olympiad. Then I noticed that it was not 1/2(a*b + c*d) but 1/2(a*c+b*d). Even then it appeared very simple.

So I made a construction assuming, W.L.G, that b>=c, where I cut a length 'c' on side BC, call the point E. Then I extended CD(intially of length 'c') to F so that CF = 'b'. Now I drew the fig in such a way that A, E, F were collinear, and hence I went onto the wrong track and joining the line AEF. Two triangle are formed whose sum of area is more than the area of quadrilateral and less than 1/2(a*c+b*d). Thus proved I thought until I gave the question to a friend. Then I found out my mistake. I was totally depressed. I kept majaar fight to try and get it in some way or other. But every construction failed.

Then a startling thing happened and I rotated the triangle ABC in the space in such a way that A and C exchange their positions. Clearly the new quadrilateral formed has same area as the older one and trivially its area was less than 1/2(a*c + b*d).

Thus proved.