2007 BMO Problems/Problem 1
(Albania) Let be a convex quadrilateral with , , and let be the intersection point of its diagonals. Prove that if and only if .
Since , , and similarly, . Since , by considering triangles we have . It follows that .
Now, by the Law of Sines,
It follows that if and only if
From these inequalities, we see that if and only if (i.e., ) or (i.e., ). But if , then triangles are congruent and , a contradiction. Thus we conclude that if and only if , Q.E.D.
Let and .
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.