<s>[INST] Square inscribed in semicircle. Find some areas. $ABCD$, $AEFG$, $GHIJ$ are squares; $ABCD$ is inscribed in the semicircle, and $E$ is the middle of $[AD]$. Attention! $JG+GA+AO \neq r$. We have to find the areas of the squares.

It's easy to find that $Area(ABCD)=\frac{4r^2}{5}$ and $Area(AEFG)=\frac{r^2}{5}$, but I can't figure out how to find the area of the last square. Can you help me? Thanks!
 [/INST] Let's let the radius of the circle be $1$ for a while and account for $r$ at the end. I'm also flipping your circle so that all of the interesting stuff happens in the first quadrant.

Consider the vertical line $x= x_i$. To build a square to the left of that with sides of length $s_i$, the upper right corner of that square will have coordinates of $(x_i+s_i, s_i)$ and will be on the unit circle. Hence
\begin{align}
   (x_i + s_i)^2 + s_i^2 &= 1 \\
   2s_i^2 + 2 x_i s_i - (1-x_i^2) &= 0 \\
   s_i &= \dfrac{-2x_i + \sqrt{4 x_i^2 + 8 - 8 x_i^2}}{4} \\
   s_i &= \dfrac{\sqrt{2 - x_i^2} - x_i}{2} \\
   x_{i+1} &= \dfrac{\sqrt{2 - x_i^2} + x_i}{2} \\
\end{align}
If we start with $x_1 = \dfrac{1}{\sqrt 5}$
Then $s_2= \dfrac{1}{\sqrt 5}$ and $x_2 = \dfrac{2}{\sqrt 5}$
$s_3= \dfrac{\sqrt 6 - 2}{2\sqrt 5}$ and $x_3 = \dfrac{\sqrt 6 + 2}{2\sqrt 5}$
$s_4= \dfrac{2+\sqrt 6 - \sqrt{30-4 \sqrt 6}}{4\sqrt 5}$ and 
$x_4 = \dfrac{2+\sqrt 6 + \sqrt{30-4 \sqrt 6}}{4\sqrt 5}$
and so on
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