I'm reading up on the derivative of $\arctan(x)$, and I understand all parts of the derivation except for the geometry section on the bottom of page 2: https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-b-implicit-differentiation-and-inverse-functions/session-15-implicit-differentiation-and-inverse-functions/MIT18_01SCF10_Ses15b.pdf
I understand we have $\tan(y) = x$. This means the ratio of the opposite side to adjacent side must be $x$, and the author chose to use the values $x$ and $1$. But why can't we use $x^2$ and $x$? It preserves the ratio ($x^2/x=x$), but it doesn't work.