Algebraic Equivalent

The original construction problems began with the Greeks, and for thousands of years, the methods were the same. Problems would be stated, a construction would be found, and then a standard geometric proof was supplied to show that the construction in fact behaved as advertised.

This worked great for thousands of years, except that it did not provide any method to show that certain constructions were impossible (see Section 6). But nobody thought that impossible constructions existed, so there was no real reason to do so.

Now we know that (in a sense) ``almost all'' constructions are impossible. In this section we will not prove that fact, but we will provide some overwhelming evidence. But to do so, we'll have to look at geometric construction from an algebraic point of view.

We will simplify the problem of construction to be that of finding only points. As we stated in Section 3.1, this is good enough, since any lines or circles that you might want can be identified in terms of a couple of points.