Are you talking about a construction using Eucliden tools? How do you inscribe a square into a given semicircle?
Here is one method . . .
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R * * * P/ | 2r
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B O Q A
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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 . . .