This page is about efficient constructions with straightedge and collapsing compass (the geometric tools of Euclid). The question of what can and cannot be constructed with this model of computation is basically answered by Galois Theory. However, given that a construction is possible, there's not much to be found on the minimum number of steps required, which incidentally is the subject of this document. 

more on the model 
The complexity of a construction is defined to be the number of drawing operations (lines and circles) that are performed. Note that other complexity measures have been proposed.
more on complexity measures 
The main feature of this web page is a Java applet in which you can view the "best" solution (that I know of) to a variety of problems. Some solutions are just amazing!
You can also use the applet to try to solve one of the problems yourself. If you happen to beat the best known solution, tell me your solution and it will replace the old one. It's an open challenge!
Choose an applet size to go to the applet page:
small applet: 580 x 320  
large applet: 850 x 520  
Euclid
paper references 