Here's the problem:
This problem could be easy, were I to know if the small pink square divided the arc length of a quarter circle into 3 pieces (identical).
What I'm trying to say is, if my guess is correct, the ratio of the length of $\dfrac{\alpha\beta}{BD}$ is $\frac13$.
But, is it correct? I want to assure myself if this hypothesis is correct. How do you prove it? This is the important key to find the small square. Because, if that's so, I can use this formula (below) to find the side of the small square:
$$\text{side} = 2r\sin\left(\frac{t}{2}\right)$$
Where $r$ is the radius of the circle and $t$ is the angle of the sector circle excribed the small square.
In conclusion, my point is just I'm asking whether it's true or not that the ratio $\dfrac{\alpha\beta}{BD}$ is $\frac13$.
Or perhaps you have another simpler way to find the small pink square?
Best Answer
The answer to your small question is yes, the arcs of the three divisions of the quarter-circles are equal
The easiest way to see this is that $\alpha$ is the same distance from $C$ as $D$ is, and the same distance from $D$ as $C$ is, because they lie on the same circles. So $\triangle \alpha CD$ is equilateral, as is $\triangle \beta BC$, and that leads to the trisection
So the ratio of the arcs $\alpha \beta:BD $ is $1:3$. But the ratio of the line segments $\alpha \beta:BD $ is not $1:3$; it is $\frac{3}{\sqrt{2}}(\sqrt{3}-1):3\sqrt{2}$ or $\frac12(\sqrt{3}-1):1$, about $0.366:1$.