Note. These demos require a Java plugin and may download a 200K library from Geometer's Sketchpad. You may have to adjust Java security settings to run them.
Simson line. I find this proof much more engaging than visualizations of static Euclidean proofs of this theorem, which abound on the Internet.
Existence of the regular icosahedron. This short modern proof told to me by Peter Doyle stands in dramatic contrast to the intricate argument in the culminating book of Euclid's Elements,