I was reading an interesting proof (published by Apostol I believe) of irrationality of $\sqrt{2}$ that aimed to use purely geometric methods to prove the result.
Although the proof is interesting on its own, I was wondering if it is possible to use a similar technique for a general argument that the square root of any non-square $k$ is irrational. I know the Apostol paper claims this can be done for $\sqrt{n^2 + 1}$ and $\sqrt{n^2 - 1}$ but the paper I read this proof in claims it can be generalised to any non-square $k$:
This proof is a prime example of the cooperation of two different fields of mathematics. We just translated a purely number-theoretical problem into a problem about triangle similarity, and used our result there to solve our original problem. This technique is widely used all over higher-level mathematics, even between things as seemingly unrelated as topological curves and groups. Finally, we leave it as an exercise to the reader to extend this proof to a proof that whenever $k$ is not a perfect square, then $\sqrt{k}$ is irrational. The proof is quite similar, but strays from nice isosceles right triangles
Is this possible? If so, I would love to see how!