I wanted to prove the Pythagorean theorem by the method of dimensional analysis. I assumed a triangle like this ($AD=h, AB=a, AC=b, DB=x, DC=y$).
Then I wrote these:
$$\frac xh + \frac yh = f(\frac ah , \frac bh)$$ $$\frac ah = f(1 , \frac xh)$$ $$\frac bh = f(1 , \frac yh)$$
Normal quetions that arise at this stage are:
Can the function $f(\alpha,\beta)$ be expressed by combination of mathematical functions?
Is the general form of $f(\alpha,\beta)$ possible to determine?
Can we take another step and find the explicit form?
Here, of course, we know the answer. What I want is the way to achieve it.