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 2 of 5: Compute the four apices

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In plain words

Once one equilateral apex is known, the rest follow by symmetry.

k=\frac{\sqrt{3}-1}{2}(-1-i),\quad l=ik,\quad m=i^2k,\quad n=i^3k

Detailed analysis

Rotating one side by $60^\circ$ toward the center gives $k=\frac{\sqrt{3}-1}{2}(-1-i)$ . The quarter-turn symmetry gives $l=ik$ , $m=i^2k$ , and $n=i^3k$ .