Consider triangles AOB and COD:
AO = CO (according to the condition, point O is the middle of the AC);
DO = BO (by condition, point O is the middle of BD);
The angles AOB and COD are vertical, which means they are equal.
Thus, triangles AOB and COD are equal according to the first sign of equality of triangles (along two sides and the angle between them), which means that these triangles have the corresponding elements, that is, side AB = CD.
Since, according to the condition of the problem, AB = 12 cm, then CD = 12 cm.
Answer: 12 cm.
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