It is very easy to construct one of the squares inscribed in a circle with the help of a ruler and a compass.
My question:
Given a circle with its center. Construct the vertices of one of the squares inscribed in this circle using ONLY compass!
I have no idea! Of course, I know the basic things, like I can reflect any point to the center, but how to continue? Thanks in advance!
Let $O$ be the center of the given circle of radius $r$. By using the compass at the same width $r$, construct the six vertices $A$, $B$, $C$, $D$, $E$, and $F$ of the regular hexagon inscribed in the circle. Then bisect the arc between two consecutive vertices $C$ and $B$ of the hexagon:
1) Draw two circles with radius $|AC|$, whose length is $\sqrt{3}r$, one centered at $A$ and another at $D$ and we obtain the intersection point $G$.
2) Draw a circle with radius $|OG|$, whose length is $\sqrt{2}r$, centered at $A$. Let $H$ and $I$ be the intersection points with the original circle.
Then $AHDI$ is desired inscribed square.
P.S. The fact that every ruler-and-compass construction could be obtained with a compass alone is due to Lorenzo Mascheroni (1750-1800).