Math, traditional subjects    Synthetic Geometry 	   Constructions via
UpFolderBtn
HomeMathGeometrySynthetic Geometry

Without constructions of some sort, whether by straightedge and compass, or by more complicated means such as linkages, the axioms of geometry are for all practical purposes empty statements. They assert the existence of an obect without saying how it is to be produced.

A long tradition of rather compulsive thinking, starting with the Greek tradition, has made something sacred out of ruler and compass constructions. It is a fascinating fact that, as Gauss first showed, objects constructed by these means correspond to algebra over the “quadratic field” – extracting square roots is the most complicated algebra we can perform. But there is really nothing essentially “purer” about limiting our constructions to those with ruler and compass as the only tools.

A compass can be as simple as a rod anchored at one end, with a drawing tip at the other end.

item3

A pair of angle bisectors isn't much more complicated. We just need equal-length cross-pieces to keep the middle piece centered.
item4

To interactively work with these examples, just click the icon below. You will also need a copy of Cinderella, available free at their website

item11

The green cross-pieces don't have to be parallel to the sides. We just need to ensure that we get the same angles on each side of the orange middle bar.
item7

Also, we don't even need the cross-pieces to be the same length. All that matters is that we have similar triangles on each side of the orange bar in the middle.

item8 item9

 Synthetic Geometry 
 Synthetic Geometry 
 Analytic Geometry 
 Projective geometry 
 Differential geometry 
 Math TOC 
 Topology 
 Calculus & Analysis 
 Home 
 Math