In Washington's book on elliptic curves, there is an exercise:
Let $n$ be an integer. Show that if $x_0, y_0$ are rational numbers satisfying $y_0^2 = x_0^3 − n^2x_0$, and $x_0 \neq 0, ±n$, then the tangent line to this curve at $(x_0, y_0)$ intersects the curve in a point $(x_1, y_1)$ such that $x_1, x_1 − n, x_1 + n$ are squares of rational numbers.
Well, implicit differentiation tells us that at $(x_0,y_0)$, the slope of the tangent line is $$y' = \frac{3x_0^2-n^2}{2y_0}.$$ So, the equation defining the line tangent to the curve at $(x_0,y_0)$ is $$y = y'x + (y_0- y'x_0).$$
If we let $u := y_0- y'x_0$ then we have $(y'x+u)^2=x^3-n^3x$ for points $(x,y)$ common to the tangent line and the curve. This leads to $0 = x^3 - (y')^2x^2-(n^2+2y'u)x-u^2$. Since $(x_0,y_0)$ is a point where the line is tangent, $x_0$ is a double root of this cubic and since the third point of intersection is $(x_1,y_1)$, we have that $u^2 = x_1x_0^2$ since the product of roots is the constant term. Since $x_0 \neq 0$ and both $u$ and $x_0$ are rational, it follows that $x_1$ is the square of a rational number.
Now, I am stuck on showing that $x_1 + n$ and $x_1 - n$ are squares of rational numbers. I've looked at the coefficient of $x^2$ in the cubic mentioned above but didn't get anywhere with that and I have tried substituting $x_1 = \frac{u^2}{x_0^2}$ into various places but couldn't find anything from this as well. Perhaps I am missing something obvious but any assistance in completing this exercise is greatly appreciated!
