There are already several versions of a geometric proof by infinite descent so maybe another isn't needed. BUT I did not see this version( maybe I overlooked it...) which is especially visually clear to people who are less scared by rectangles than by triangles (I'll describe rather than draw the diagram): Take a rectangle in proportions 1 to sqrt(2) (like an ideal sheet of A4 paper). Remove a largest square from the top leaving one with proportions 1 to sqrt(2)-1 along the base. Then remove a largest possible square from the right side of this smaller rectangle. The result is an even smaller rectangle proportional to the first. SO (as usual) the Euclidean algorithm never ceases.