Find all integer points!

Drop a perpendicular. Then $\tan(\alpha) = \frac{y_0}{1+x_0}, \tan(\beta) = \frac{y_0}{2-x_0}.$ Since $\tan(\beta) = \frac{2\tan(\alpha)}{1-\tan^2(\alpha)},$ we get $\begin{aligned} \frac{y_0}{2-x_0} &= \frac{2\frac{y_0}{1+x_0}}{1-\frac{y_0^2}{(1+x_0)^2}} \\ \frac{y_0}{2-x_0} &= \frac{2y_0(1+x_0)}{(1+x_0)^2-y_0^2} \\ \frac1{2-x_0} &= \frac{2(1+x_0)}{(1+x_0)^2-y_0^2} \\ 2(2-x_0)(1+x_0) &= (1+x_0)^2-y_0^2 \\ y_0^2 &= (1+x_0)(1+x_0-2(2-x_0)) \\ y_0^2 &= (1+x_0)(3x_0-3) \\ y_0^2 -3x_0^2 &= -3 \end{aligned}$

This is related to Pell's equation . The solutions are of the form $y_0+x_0\sqrt{3} = \sqrt{3}(2+\sqrt{3})^n$ for positive values of $n.$ This leads to the solutions $(2,3), (7,12), (26,45), (97,168), (362,627), (1351,2340), (5042,8733), \ldots$ There are $\fbox{6}$ with $x$ -coordinate in $(1,2020].$