Monday, March 01, 2010

Cute "proof" of Pythagoras' theorem

[For reasons I have yet to fathom, the ASCIIMathML plugin doesn't want to work for this page...]

I'm reading The Mathematical Mechanic by Mark Levi, a tour of various mathematical results that can be obtained much more directly (albeit without proof) by appealing to our mechanical intuition.

Here's a beautiful proof from the book of Pythagoras' theorem, a^2 + b^2 = c^2 where a and b are the opposite and adjacent sides of a right angled triangle and c is the hypotenuse.

Consider an ice skater of mass m on a perfectly smooth ice rink. The skater starts in the South West corner of the rink and pushes off against the South wall; the skater is now moving North with velocity a and hence has kinetic energy ma^2/2. Next the skater pushes off against the West wall and adds an Easterly component b to their velocity. The energy acquired from this second push is mb^2/2. Now, the skater's overall velocity is c and overall kinetic energy is mc^2/2. But this must be the sum of the energy acquired from each push, hence ma^2/2 + mb^2/2 = mc^2/2. Cancel the m/2 terms and you have Pythagoras' theorem!