Dividing a square into three equal parts, using ruler and compass only

Question

Is it possible to divide a square into three parts of equal area, using ruler and compass only? If so, how do you do it?

Answer 1

Yes, it's doable. Consider the illustration below for side $AB$ of your square,

where you draw circles of radius $r=\overline{AB}$ centered at both $A$ and $B$ and connect their crossing points $E$ and $E'$, yielding the midpoint of segment $\overline{AB}.$

Then draw three circles of radius $r_2=\frac{\overline{AB}}{2}$, centered at $A$,$B$ and the midpoint from above. Connect the top crossing point $E$ with $C$ and $D$, the two bottom crossing points of the smaller circles. This trisects segment $\overline{AB}$ into $\overline{AP_1}$, $\overline{P_1P_2}$ and $\overline{P_2B}.$

Repeat for segment $\overline{CD}$ in your square $ABCD$, and connect the trisection points $P_1P_3$ and $P_2P_4.$ Done.

Answer 2

Suppose you know how to draw a line perpendicular to another line through a point on it (classic problems you shall be able to solve in order to use the as "lemmas"), just easily go through the steps here, you can mark all the important points on the sides using your ruler or compass. (I've done so and could solve the problem.) Hope this helps you :)