BY-MO 1999 Problem 4

A circle is inscribed in the trapezoid ABCD. Let K, L, M, N be the points of tangency of this circle with the diagonals AC and BD, respectively (K is between A and L, and M is between B and N). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$ , find the radius of the circle.

[Moderator edit: A solution of this problem can be found on http://www.ajorza.org/math/mathfiles/scans/belarus.pdf , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.]