I have a minor personal triumph to relate today. So, I try to have several cute, quick math puzzles/facts on hand to bust out at parties, because I am cool like that. The most popular of these is a participation-based trick that goes as follows (feel free to play along at home):

Take a pen and paper and draw a quadrilateral. There are no restrictions (it can be concave or self-intersecting), but don’t make it too close to the sides of the paper. Now, for each edge, draw the square containing that edge that is outside the quadrilateral. Put a dot in the center of each of the four squares, and draw a line connecting opposite dots, ie, those that came from opposite edges.

The Punchline: The lines you just drew are the same length, and perpendicular.

(If you lack pen and paper, theres an applet here)

It works pretty well because it isn’t very sensitive to sloppy geometry on the part of the artist; so you will pretty consistantly get ‘perpendicular-looking’ lines. Also, I often bend the truth a bit and attribute it to Napolean Bonaparte, even though he proved something different but closely related. The usual attribution of *this* result is to van Aubel, who to my knowledge conquered very little of Europe.

I’ve been using this for a couple of years now, and periodically I attempt to find a nice geometric proof without passing to coordinates. Such a proof eluded me up to last night, when I came up with a reasonably nice vector-based proof. Its a little cheap since using vectors isn’t totally different from passing to coordinates, but in my mind a geometric proof is one which can be done only with pictures (though I will use words for laziness’ sake). For the record, I was unaware of van Aubel’s proof until this morning, which is a more traditional geometric proof, but a little bit more indirect.