 Euler theorem on four squares in arithmetic progression

There is no nonconstant four-term arithmetic progression of rational square numbers. If $r^2,s^2,1,t^2$ formed one, then $(-2s^2,2rst)$ would lie on $y^2=x(x+1)(x+4)$; its explicitly known Mordell-Weil group forces $s^2=1$ and the common difference to vanish.