The chords AB and CD meet at point K. Find DK: BK if AK: CK = 3: 2.

Note that angles ACD and ABD rest on the same arc AD.

This means that the angles ACD and ABD are equal.

Also note that the angles BAC and BDC rest on the same arc BC. This means that the angles BAC and BDC are equal.

Consider triangles ACK and BDK.

We have proved that the angles ACK = KBD and KAC = BDK.

Therefore, triangles ACK and BDK are similar in two angles.

Hence, the ratios of the lengths of the sides lying opposite equal angles are equal to each other. Hence, we have:

DK / BK = AK / CK = 3/2.

Answer: DK / BK = 3/2.