The 3 Ancient Construction Problems

Problems about construction using compass and straightedge preoccupied mathematicians for 2000 years. The first three postulates of Euclid’s Elements (300 BC) involve constructions and the first proposition was about the construction of an equilateral triangle. Constructions of other regular polygons were solved, but the heptadecagon (17-gon) proved to be difficult. It was only solved by Carl Friedrich Gauss in 1796 when he was just 19 (see the following animation).

17-gon construction

Aside from the heptadecagon problem and other construction problems, numerous mathematicians attempted to solve the three most difficult ancient construction problems: trisection of any given angle, duplication of a cube, and squaring the circle. » Read more

Introduction to Geometric Construction

In Geometry, the term construction refers to a precise way of drawing using two tools, an unmarked straightedge and a compass. The straightedge is assumed to have an infinite length and has no markings.  In constructions, these tools and only these tools must be used to construct lines, curves, and polygons.  Shown below is an animation showing  how to construct a square using compass and straightedge.

Square construction

The Construction Rules

In construction, aside from the tools, there are specific rules to be followed. These rules are the following.  » Read more

GeoGebra Essentials 5 – The Compass Tool and the SSS Congruence

This is the fifth tutorial in the GeoGebra Essentials Series. If you are not familiar with GeoGebra, you may want to read the Introduction to GeoGebra post and prior tutorials. They contain the pre-requisites of this tutorial.

In the tutorial below, menu commands, located in the menu bar, are in brown bold text, and submenus are denoted by the > symbol. For example, Options>Labeling> New Points Only means, click the Options menu, choose Labeling from the list, then select New Points Only. The tool texts are colored orange. For example, New Point means the new point tool.

geogebra - triangle congruence

In this tutorial, we will mimic compass and straightedge construction using GeoGebra’s Compass tool, Segment and Ray tools. We use the concept of the SSS congruence1 to construct a triangle congruent to a given triangle. » Read more