CN-LEAGUE 2012 Problem 11

In the Cartesian plane $XOY$ , there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$ , where $O$ is the origin.(1) Prove that $|OA|\cdot |OB|$ is a constant.(2) Find the locus of $C$ if $A$ is a point on the semicircle $$(x-2)^2+y^2=4 (2\le x\le 4).$$