Compass and straightedge constructions
With only a compass and a straightedge that has no markings, what can you draw exactly, and how do you know it is exact?
Where you are 8 inside
What this area is about
A construction is a drawing made with two tools: a straightedge, which draws the line through two points and cannot measure, and a compass, which draws circles and holds one width. Nothing is read off a scale, so every step can be justified by a definition or a theorem, and the result is exact for every starting figure, not just the one on the page.
How these fit together
The basic moves. Copying a segment and an angle carries a length or an angle to a new place. Bisecting a segment and an angle finds a midpoint and splits an angle in half.
Building on them. Perpendicular and parallel lines through a point, constructing a triangle from given parts, the equilateral triangle, square and hexagon in a circle, constructions with circles and dividing a segment into n equal parts are procedures that reuse the basic moves.
Why they work. Why a construction works is the concept behind all of them: each step creates equal distances or one fixed line, and a proof, often through congruent triangles, turns those into the claimed property.
A path through it
Copying comes first, before bisecting and before constructing a triangle from given parts. Bisecting comes before perpendicular and parallel lines, constructions with circles, and why a construction works. Perpendicular and parallel lines come before the equilateral triangle, square and hexagon, and before dividing a segment into equal parts.