Construction Versus Proof

This flow traces the tension between drawing a requested figure and demonstrating that the drawing satisfies its intended definition.

increasingBearing stren­gth in each eventdecreasing

Requested Figure

Requested Figure Construction Goal The first problem asks for an equilateral triangle on a given finite line. The task is practical and visual. Unsettled Status A drawing alone cannot yet establish that the three sides are equal.

Instrumental Action

Instrumental Action Circle Intersections Circles are described from the endpoints of the given line with the line as radius. Their intersection supplies the third vertex. Constraint Every visible alteration is authorized by a postulate.

Proof of Equality

Proof of Equality Radius Relations The new sides are radii of circles whose radii equal the original line. The definition of the circle therefore proves all three sides equal. Transition The constructed figure becomes a theorem only after this reasoning.

Perpendicular Construction

Perpendicular Construction Two Cases A perpendicular is drawn either through a point on a line or from a point outside it. The constructions differ because their available positions differ. Verification Congruent triangles establish the right angle rather than relying on visual appearance.

Constructive Certainty

Constructive Certainty Ratios Made Visible Third fourth and mean proportionals are constructed through parallels and semicircles. The figures embody relations that might otherwise remain symbolic. Resolution Construction and proof complete one another: making provides the object, reasoning certifies it.

at the first problem of Book I

during the first circle construction

at the close of the first proposition

at the eleventh and twelfth propositions of Book I

at the construction propositions of Book VI