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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 | |

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

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

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

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. | ||||

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.
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