Geometric Constructions
After the basic geometric shapes, more elaborate objects were subjected to the action of the ruler and compass. Can we obtain ellipses or parabolas point by point and reconstruct all their notable points? We enter the universe of constructible numbers... or not. Finding which ones are is quite an art. The power of the techniques developed has invaded number theory and made it possible to resolve — in the negative — a problem more than two thousand years old: the quadrature of the circle!
All articles in this folder
The ellipse and its many features
Although descriptive geometry provides laborious methods for constructing an ellipse with straightedge and compass (see the article "Albrecht Dürer and technical drawing"), other, more conventional methods also exist. A point on the curve, the center, foci, axes and tangents: let's take a quick tour of the available techniques!
The parabola in all its finery | Tangente
A common exercise in the days of descriptive geometry was to construct a point on a curve and its tangent. Following the ellipse, here is a short survival guide offering a handful of constructions for solving the same problem for the parabola.
From geometry to algebra: constructing numbers
Starting with two distinct points in the plane and performing only finitely many ruler-and-compass constructions, can we obtain any point in the plane? If not, which points can we obtain in this way?
Triangles with an angle trisector that is a median
A geometric question about trisecting an angle can lead to some astonishing discoveries! Along the way, we encounter triangles, angle trisectors that are also medians, the non-constructible 20° angle—and curves not yet catalogued?
Straightedge and compass in education
How long have schoolchildren been drawing geometric figures with a straightedge and compass? More importantly, why? Even elementary geometric constructions must be introduced progressively, and this requires knowledge to be organized systematically.
Why a sound method matters in geometry | Tangente
Never known how to "get started" on a straightedge-and-compass geometry problem? What matters, which points should you introduce, and which line should you construct? The classic, tried-and-tested method of analysis and synthesis provides a sound approach to such construction problems.