In the interior of a square
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
we construct the equilateral triangles
![ABK, BCL, CDM, DAN.](/media/m/6/4/7/647fdc5eaa16a4dd51480b80e001533a.png)
Prove that the midpoints of the four segments
![KL, LM, MN, NK](/media/m/a/6/6/a66b170da0c07065a559b17ef9839374.png)
and the midpoints of the eight segments
![AK, BK, BL, CL, CM, DM, DN, AN](/media/m/e/f/5/ef5be8abb767c87cf3697fb2fa26cad1.png)
are the 12 vertices of a regular dodecagon.
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In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.