Points M, N, P, Q lie on the sides AD, AB, BC, CD of parallelogram ABCD

Points M, N, P, Q lie on the sides AD, AB, BC, CD of parallelogram ABCD, respectively, so AM / AD = AN / AB = PC / BC = CQ / CD = 1/3. Prove that MNPQ is a parallelogram.

By condition, AH / AB = 1/3, CQ / SD = 1/3, and since AB = SD as opposite sides of the parallelogram, then AH = CQ.

Similarly, AM = CP.

Angle HAM = РСQ, then triangles АНМ and РСQ are equal on two sides and the angle between them. Then НМ = РQ.

Similarly, triangles HBP and MDQ are equal, then PH = MQ.

In the quadrangle МНРQ, the opposite sides are pairwise equal, hence it is a parallelogram, which is what we had to prove.