Lesson 4 of 20 · 7 moves

The Square

Build a square on the segment AB.

The finished construction: the figure in gold over the lines that found it.

The square is the symbol of earth and of measure: four equal sides, four right angles. To raise it you need a right angle at B.

Hint of the masons: extend AB past B by its own length, then bisect the doubled segment. The bisector stands straight up from B.

The app snaps to every crossing, checks your construction exactly, and offers hints one move at a time.

Step by step

Bisect draws the perpendicular bisector of two points in one move: it stands for the circles and line of the perpendicular bisector.

Extend AB into a full line.
Circle centred on B through A: it meets the line again beyond B.
Bisect A and that new point: the bisector rises straight up from B.
Circle centred on A through B.
The first circle met the upright at C. Centre a circle on C through B: it finds D.
Join C to D.
Join D to A to close the square.

Earth, measure, the four directions. Its diagonal is √2 times its side, the first number the Greeks proved could never be a fraction.