Hello, Garas!
Construct the line segment which is a geometric mean of two given line segments. Code:
S * * *
* *
*  *
*  x *

*  M *
**+*
P a Q b R
Measure and consecutively on a line.
Bisect the line and locate its midpoint .
Use the as center and as radius,
. . and draw a semicircle.
At , erect a perpendicular meeting the semicircle at
Then is the geometric mean of and
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Why would we need such a construction?
Suppose we have an rectangle
. . and we need a square with the same area.
Code:
**
**  
    x
b    
   
** **
a x
Then: .
Edit: too slow (again) . . . *sigh*
.