Henry Ernest Dudeney/Modern Puzzles/127 - With Compasses Only/Solution

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Modern Puzzles by Henry Ernest Dudeney: $127$

With Compasses Only
Can you show how to mark off the four corners of a square, using the compasses only?


Solution

The solution given by Dudeney makes the assumption that the compass is stiff enough that when you lift it from the paper, you can put the point down and make another circle with the same opening that you used for the last one.

Dudeney-Modern-Puzzles-127-solution.png

Draw a circle $OA$.

Then, using the same radius $OA$, make:

the arc at $B$ with center $A$
the arc at $C$ with center $B$
the arc at $D$ with center $C$.

Then make:

an arc at $E$ with center $A$ and radius $AC$
another arc at $E$ with center $D$ and radius $DC$.

$OE$ is now the length of one side of the square required.

Now, using the length $OE$ as the radius, draw the arcs cutting the original circle at $F$ and $G$.

The square now has vertices $AFDG$.


Some may consider the solution flawed, as you have to draw the arcs with radius $OE$ at a point different from both $O$ and $E$.


You can rescue this by constructing another circle whose center is at $D$ and whose radius is $O$, and constructing a point in the same position relative to it as $E$ is to circle $OA$.

Then one of the arcs used to construct that point intersects the circle $OA$ at $F$, and the job is accomplished thus.


Sources

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