Inscribe Square in Circle using Compasses Alone/Corollary

Problem

To inscribe a regular dodecagon in a given circle,

by means of compass alone,

supposing the center to be known.

Solution

We use the same construction as for Inscribe Square in Circle using Compasses Alone:

Then we note that, for example, $C$ and $E$ are $30 \degrees$ apart.

Hence they are adjacent vertices of a regular dodecagon inscribed in the circle.

It remains to construct the remaining vertices, which is done by, for example, setting the compass to $CE$ and drawing an arc with center $C$ and working round the circle.

$\blacksquare$

Sources

• 1821: John Jackson: Rational Amusement for Winter Evenings
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Rational Amusements for Winter Evenings: $158$