Thanks!

2. Re: Circumscribe a square about a semicircle!

Hello, Jammix!

I suspect that I know the solution . . . but it's only a guess.

Circumscribe a square about a semicircle.

Code:
                R
B * - -.-.*.o.*.-.- * C
|  .*:::::|:::::*.|
| *:::::::|:::::::o S
|*::::::::|:::::* |
|:::::::::|:::* r |
*:::::::::|:*     *
Q o:-:-:-:-:O - - - o
*:::::::* :       *
|:::::*r  :       |
| ::*     :       |
A * o - - - * - - - * D
P       E
Square $ABCD$ is circumscribed about the (tilted) semicircle.
The circle has center $O$ and radius $r\!:\;OP = OQ = OR = OS = r.$

$\Delta OEA$ is an isosceles right triangle.
Hypotenuse $PO = r \quad\Rightarrow\quad OE = \tfrac{\sqrt{2}}{2}r$

Let $x = AB$, the side of the square.

Then: . $x \:=\:RO + OE \:=\:r + \tfrac{\sqrt{2}}{2}r$

Therefore: . $x \:=\:\left(\tfrac{2 + \sqrt{2}}{2}\right)r$

3. Re: Circumscribe a square about a semicircle!

Can you give me the steps in which you constructed that? Thanks!

4. Re: Circumscribe a square about a semicircle!

Correction: . $\Delta OE\color{red}{P}$ is an isosceles right triangle.

(I was not allowed to edit my post. Why?)