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 4 of 5: Midpoints on triangle edges

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In plain words

Two representative computations plus rotational symmetry account for all eight edge midpoints.

\left|\frac{1+k}{2}\right|=\left|\frac{i+k}{2}\right|=\frac{\sqrt{3}-1}{2}

Detailed analysis

Direct calculation gives arguments $-30^\circ$ and $120^\circ$ for the midpoints of $AK$ and $BK$ . Rotating by $90^\circ$ supplies the other six edge midpoints, at the remaining odd multiples of $30^\circ$ .