What should have been a simple exercise in geometry has morphed into a multi-day affair with me figuratively tearing my hair out. I have no clue what's wrong.
This image accompanies the problem:
The problem is "simply" to show that $$r=\frac{\rho}{1+\frac{\rho^2}{4L^2}}$$ I can do the calculus and differential geometry that follows, but I cannot figure this out. I've even resorted to getting out a ruler and measuring the relevant quantities to check the answer.
Taking this head-on has done nothing, so I tried to reverse engineer the solution. Calling the large hypotenuse $x$, we have $$\rho^2+4L^2=x^2$$ Inserting this in the answer, we have $$r=\frac{4L^2\rho}{x^2}$$ I think this is about all that I've been able to to productively.
A huge thank you to anyone who helps in any capacity.
This is a problem involving similar triangles. Can you see the related ratios: $$\frac{L-\sqrt{L^2-r^2}}{r}=\frac{2 L}{\rho }?$$