MathLabs

Problem 1

Inside square ABCDABCD, construct equilateral triangles ABKABK, BCLBCL, CDMCDM, and DANDAN. Prove that the midpoints of KL,LM,MN,NKKL,LM,MN,NK and of AK,BK,BL,CL,CM,DM,DN,ANAK,BK,BL,CL,CM,DM,DN,AN are the twelve vertices of a regular dodecagon.
Step 5 of 5: Identify the twelve vertices
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In plain words

Equal radius and equal angular gaps are precisely the definition of a regular dodecagon.

zj=3−12eijπ/6,j=0,1,…,11z_j=\frac{\sqrt{3}-1}{2}e^{ij\pi/6},\quad j=0,1,\ldots,11
The unit circle displays the 30∘30^\circ angular spacing of the twelve midpoint directions.
A unit circle with a marked point at angle $30^\circ$, representing consecutive dodecagon directions.
Detailed analysis

The twelve midpoint arguments are exactly jπ/6j\pi/6 for j=0,…,11j=0,\ldots,11, and every modulus is 3−12\frac{\sqrt{3}-1}{2}. Hence they are equally spaced on one circle.