(1) 解:因为$OA = OB,$$OC\perp AB,$所以$\angle BOC=\angle AOC = 2\angle ADC。$因为$\angle ADC = 30^{\circ},$所以$\angle BOC=60^{\circ}。$
(2) 证明:因为$\angle AOC=\angle BOC,$所以$AC = BC。$因为$\angle BOC=60^{\circ},$$OB = OC,$所以$\triangle BOC$是等边三角形,所以$OB = BC。$因为$OA = OB,$所以$OA = OB = BC = AC,$所以四边形$AOBC$是菱形。