Circle measurement of Archimedes Let $f_n$ or $F_n$ be areas of the regular $n-$ polygon described to the unit circle or circumscribed. Show 
$f_{2n}=\sqrt{f_nF_n}$ and $F_{2n}=\frac{2f_{2n}F_{n}}{f_{2n}+F_n}$
In the solutions there is only the fact that this follows from elementary geometric considerations.
I tried to explain the result by elementary geometry, but I did not succeed.
I have already searched the net for approaches. For example you can construct a $2n$ corner out of a $n-$ corner which is in a circle by drawing a point between two corners of a side.
To clarify what I mean I have added a sketch using the example of a 4 corner. 
Why is the geometric mean of the area content of the outer and inner 4 corners now the area content of the inner $8$ corner?
I have no approach to the formula for the outer $2n$ corner.
How can I draw the $2n$ corner from the outer $n$ and why is the harmonic mean of the area of the inner $2n$ corner and the area of the inner $n$ corner?
I hope someone can help me.


The key idea is that a regular $n$ sided polygon is composed of $n$ isoceles triangles each having a common vertex at the center of the polygon. Also, each isoceles triangle is split by an altitude into two congruent right triangles. Thus, the area of the polygon is $2n$ times $T$ the area of one of the right trangles whose vertex angle is $t:=\pi/n$. Given a circle of radius $1$ there are two cases. If the polygon is inscribed inside the circle, then $T = \frac12 \sin(t)\cos(t).\,$ If the polygon is circumscribed outside the circle, then $T = \frac12 \sin(t)/\cos(t).$
Putting this together, we have $\,f_n = n\sin(\pi/n)\cos(\pi/n)\,$ and $\,F_n = n\sin(\pi/n)/\cos(\pi/n).\,$ The relations between $\,f_n,F_n\,$ and $\,f_{2n},F_{2n}\,$ is a consequence of simple trigonometric identities such as
 $\,1 = \sin^2(t)+\cos^2(t),\,$ but these are based on geometric relations such as the Pythagorean theorem. You also need the doubling formulas $\,\sin(2x)=2\cos(x)\sin(x),\,\,\cos(2x)=\cos^2(x)-\sin^2(x).$
Note that Archimedes used the perimeters of polygons and not their areas to approximate $\pi$.
