Quadrilateral ABCD is a parallelogram, segments AL, BN, CM and DK are equal. Prove that LNMK is also a parallelogram.

Since, by condition, NB = MC = DR = AL, then both AN = CR and BM = DL, since the opposite sides of the parallelogram are equal, and segments of equal length are subtracted from them.
Opposite angles of a parallelogram are equal, angle NAL = MCR, angle NBM = RDL. Then triangle NBM is equal to triangle LDR, and triangle RCM is equal to triangle LAN along two equal sides and the angle between them. Then NM = RL and NL = MR.
If the opposite sides of a quadrilateral are equal, then such a quadrilateral is a parallelogram, as required.