Hello, Pabo258!
How do you inscribe a square into a given semicircle? Are you talking about a construction using Eucliden tools?
Here is one method . . . Code:
C * -
/| :
/ | :
/ | :
R * * * P/ | 2r
* - - - - o | :
* : /: * | :
* : / : *| :
: / : | :
* : / : * :
*----+----*----*----* -
B O Q A
: - r - : - r - :
We have a semicircle with center $\displaystyle O$ and radius $\displaystyle OA = OB = r.$
At $\displaystyle A$ erect a perpendicular to diameter $\displaystyle AB$
. . and measure off a distance $\displaystyle 2r\!:\;AC = 2r$
Draw line segment $\displaystyle OC$, intersecting the semicircle at $\displaystyle P.$
From $\displaystyle P$ drop a perpendicular $\displaystyle PQ$ to diameter $\displaystyle AB.$
From $\displaystyle P$ draw a horizotal line, intersecting the semicircle at $\displaystyle R.$
Segments $\displaystyle PQ$ and $\displaystyle PR$ determine the inscribed square.
I'll let someone else provide the proof . . .