Inscribe Square in Circle using Compasses Alone

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Problem

To inscribe a square in a given circle,
by means of compass alone,
supposing the center to be known.


Corollary

To inscribe a regular dodecagon in a given circle,
by means of compass alone,
supposing the center to be known.


Solution

Let $K$ denote the circle.

Inscribe-Square-in-Circle-with-Compasses.png

Locate point $B$ on the circumference at which one of the vertices of the square is to be located.

Construct $X$, $C$, $A$ from arcs of radius of $K$ center $B$, $X$ and $C$ respectively.

Construct arcs $CD$ and $XD$ with center $B$ and $A$ respectively of radius $BC$ and $AX$ respectively.

Construct $E$ and $F$ on the circumference of $K$ so as to make $AE$ and $AF$ the same length as $OD$.


This article, or a section of it, needs explaining.
In particular: The problem with this method of construction is that in a conventional compass and straightedge construction, you are not allowed to set the spread of the compass based on an existing length (in this case $OD$), and then pick up the compass and place the pointy end down at a different point, and keep the same spread. Once you have removed the compass from the page, it is assumed that the compass automatically closes. As this is assumed to be the same sort of compass as that, you can't "just" pick up the compass and move it and keep the same spread. It needs to be demonstrated that there is such a construction. I believe there is, because of the as-yet undocumented Mohr-Mascheroni Theorem, so that it is in theory possible -- but I believe that Jackson (and by extension Wells) has glossed over that detail, not having noticed the subtlety.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
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$\Box AEBF$ is the square required.

$\blacksquare$


Sources

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