CHEERFUL FACT To see a given solution lay an mxn sheet of paper on the desk A_____ m ____ B
n n
C_____m_____D
Fold and crease so that corner D falls on corner A. The triangle bounded by AB and the image of BD is (up to similarity) T(m,n).
this is not a coincidence.
I agree. Here's a less sophisticated argument. A_____ _ ____ B | | | | P | | | | | | n | | | | C_____m_____| D let P be the point on BD which is one of the endpoints of the diagonal crease, so that ABP is the right triangle of interest. Let x be the length BP. Then AP(the hypotenuse}=n-x so (n-x)^2=x^2+m^2, and solving for x gives the result. I was serious when I said this was a typical 9th grade algebra problem where the main thing you learn is that you're supposed to let x be equal to something. The kids have probably seen the 3,4,5 triangle and you show them 5,12,13, (homework: check that it works) and say there are infinitely many other such magic triples, and let them start folding the paper. dg