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 and radius
At erect a perpendicular to diameter
. . and measure off a distance
Draw line segment , intersecting the semicircle at
From drop a perpendicular to diameter
From draw a horizotal line, intersecting the semicircle at
Segments and determine the inscribed square.
I'll let someone else provide the proof . . .