Constructions

What you can build with compass and straightedge alone.

The idea

A classical construction draws a figure with exactly two tools: a compass, which draws a circle of chosen radius about a chosen point, and an unmarked straightedge, which draws the line through two given points. Neither tool measures anything — there is no ruler and no protractor — so a construction must be exact by argument rather than by reading a scale.

The two tools suffice because a circle records distance: every point on it lies at the same distance from the center. An arc drawn about a point therefore marks all the places at one chosen distance from that point, and where two arcs cross lies a point whose distances to both centers are known. Nearly every construction is assembled from this one move, and its proof consists of naming the equal lengths the arcs created.

Each construction is therefore a fixed recipe with a justification. Some tasks admit no recipe at all: the two tools generate only certain lengths from the given ones, and a task requiring a length outside that supply is impossible, not merely difficult.

The figure shows the model construction: equal-radius arcs about the endpoints $A$ and $B$ of a segment cross at two points, and the line through the crossings bisects $AB$.

Ways to work on it

• Walkthrough. The classic recipes: bisect, build, and copy.
• Practice. Match a construction to its method.
• Hardest. Which construction is provably impossible?

Not sure where to start? Take the ten-question placement test.