I was asked an interesting question by a friend:
You are given half a circle and a line $d$ of length $6$ is given. You build a perpendicular to the base line (the diameter of the half-circle) and complete a rectangle, such, that the top side is tangent to the circle (see an animation below). What is the area $A$ of the resultant rectangle.
So, I have a very quick question,
if I have all the data needed to calculate the area $A$?
And, since I've been asked by another human, I assume that the answer is yes. So I conclude, that the while the area is a function $f(r, d)$, where $r$ is the radius of a circle, it should be that $r = r(d)$, and in fact it follows that $A = f(d)$. Then I pick a convenient choice of a circle and calculate the area in my head. Basically, setting $d=2r$ is one such particularly easy choice, and $$A = \frac{d^2}{2}.$$
However, I have cheated, and now I need to establish somehow, that changing the circle, have no impact on the area. (I am able to check another choice of setting an angle to $45$ degrees between a diameter and $d$).
Hence, I need to make calculations (using pen and paper). On the image below, I have added $x$ to be the length of the line to the left of the perpendicular. Then by triangle similarities we have the following:
$$ \frac{d}{2r-x} = \frac{2r}{d}$$
or, more compactly, $\frac{d^2}{2} = r\cdot (2r-x)$. Which is accidentally the area.
I, personally, do not like to make heavy calculation in problems, where some simple reasoning should be applied. Therefore, I am looking for a simpler, just-in-head, solution, to why the area of the rectangle on the animation below, is independent of the circle radius?
I will attach an image and reference the reader to the Euclid's proof of Pythagorean theorem (Euclid's proof).
This answer is credited to @Intelligenti pauca.