IMO 1977 SL 12

On the sides of a square ABCD one constructs inwardly

IMO 1977 SL 12

Origin: NET

Problem

On the sides of a square ABCD one constructs inwardly equilateral triangles ABK, BCL, CDM, DAN. Prove that the midpoints of the four segments KL, LM, MN, NK, together with the midpoints of the eight segments AK, BK, BL, CL, CM, DM, DN, AN, are the 12 vertices of a regular dodecagon.