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 3 of 5: Midpoints between apices

In plain words

The four apex-edge midpoints already form a smaller square on the target circle.

M_{KL}=\frac{k+l}{2}, M_{LM}=\frac{l+m}{2}, M_{MN}=\frac{m+n}{2}, M_{NK}=\frac{n+k}{2}

Detailed analysis

Substitution gives four points with modulus $\frac{\sqrt{3}-1}{2}$ and arguments $270^\circ,0^\circ,90^\circ,180^\circ$ .