I recently discovered this theorem in geometry (which also solves a number theory problem about "Prouhet Tarry Escott" set pairs). Take an equilateral triangle with side=z. Cut it with a line segment (of length=t) from a vertex to opposite side. This divides that side into line segments of lengths x and y. Of course x+y=z. Theorem: 2*t^2 = x^2 + y^2 + z^2 and 2*t^4 = x^4 + y^4 + z^4. Picture: http://dl.dropbox.com/u/3507527/SmithPTETriangleThm.png