In 1958, E. Teuffel proved that no two hypotenuses of the spiral of Theodorus will ever coincide, regardless of how far the spiral is continued. Apparently this was proved in "Eine Eigenschaft der Quadratwurzelschnecke", Math.-Phys. Semesterber. 6 (1958), pp. 148-152. I can't identify the journal, though my hunch is it's Mathematische Semesterberichte. I haven't been able to track down a physical copy nor an electronic one through my institution. Does anybody know a proof of this fact?
To phrase it formally, the $n$th triangle has hypotenuse $r_n = \sqrt{n+1}$ and origin-angle $$\theta_n = \arctan 1/r_{n-1} = \arcsin 1/r_n = \arccos(r_{n-1}/r_n).$$ Let $\Theta_n = \theta_1 + \theta_2 + \dotsb + \theta_n$. Then, the spiral's "corners" have polar coordinates $$(r_n, \Theta_n)$$ where $n \ge 0$. Two hypotenuses coincide iff $\Theta_n \equiv \Theta_m \pmod{2\pi}$ and $m \ne n$. Is this possible?