Problem 1

Inside square

ABCD

, construct equilateral triangles

ABK

BCL

CDM

, and

DAN

. Prove that the midpoints of

KL,LM,MN,NK

and of

AK,BK,BL,CL,CM,DM,DN,AN

are the twelve vertices of a regular dodecagon.

Step 1 of 5: Choose fourth roots of unity

In plain words

Complex multiplication makes the required quarter-turn symmetry transparent.

A=1,\quad B=i,\quad C=-1,\quad D=-i

Detailed analysis

By similarity, put the square center at the origin with $A=1$ , $B=i$ , $C=-1$ , $D=-i$ . Similarities preserve midpoint ratios and angles.