ABCD is a unit square. There are 2 circles in picture. The center of one is A and passing from F. The center of another is D and passing from C.
If $\alpha=2\beta$ then I got the following results
$|AF|=x$
$\cos(2\beta)=\frac{1}{x}$
and from isoscale triangle (ADG): $\sin(3\beta)=\frac{x}{2}$
$\sin(3\beta)\cos(2\beta)=\frac{1}{2}$
$\sin(5\beta)+\sin(\beta)=1$
$\sin(\beta)=P$
my first question: Is it possible to find a solution of P via radicals?
Second question: During my drawing the figure that Area of (CGH region) is equal about Area of (HBF region) (just I noticed with geometrical drawing tool).I do not claim that they are equal Because I know very well that it is impossible to Squaring a circle.(more info is in Wiki link)
I try to find approximation of $\pi$ in radicals via the figure if possible. Note:If you know interesting $\pi$ approximations via Squaring a circle, I would like to know them. Thanks for advices and answers
EDIT:
I asked the question to wolfram alpha. I got fifth degree polinom with real coefficients.
$\sin(5\beta)+\sin(\beta)=1$
$\sin^5(\beta)+5\sin(\beta)\cos^4(\beta)-10\sin^3(\beta)\cos^2(\beta)+\sin(\beta)=1$
$\sin^5(\beta)+5\sin(\beta)(1-\sin^2(\beta))^2-10\sin^3(\beta)(1-\sin^2(\beta))+\sin(\beta)=1$
$\sin(\beta)=P$
$16P^5-20P^3+6P=1$
Really $P=\sin(\pi /6)=\frac{1}{2}$ is a solution of the polinom.
All roots are in the link. What a pity that wolfram did not offer me radical solutions for 4 degree polinom either. I will check other tools to find radical solutions of the roots. For my second question UPDATE:
I decided to see if really Areas are about eqaul to each other.
the root is for my first drawing: I will take the solution about $P\approx0.188286$ from wolfram solution.
$\sin(\beta)\approx0.188286$
$\beta\approx0.189416$
$|AF|=x$
$\cos(2\beta)=\frac{1}{x}$
$x=\frac{1}{\cos(2\beta)}\approx1.076314$
Area of (CGH region)= (wiki link)
Area of (HBF region)= (Wiki Link)
RESULT: I wonder if I use more digits of the root what Areas will be. it seems that numerical solution of areas are very near each other. I will update if I find radical solution of polinom and I will update if I find some other results. Your helps will be appreciated very much to analyze the results.
Thanks a lot to Chris K. Caldwel for his contribution. I drew the solution he offered .