Square Inscribed in a Semicircle by Jiyoon Chun
1. When the side length of a square is given as a unit
Suppose we have a square ABCD of one side is a unit length 1.
2. Geometric mean by similarity of triangles.
Constructing Inscribed Square when the Semicircle is given
As I showed at 1), it is easy to construct circumscribed semicircle when a square is given. If we duplicate the square, we can easily get the diameter of the circumscribed circle of the rectangle. However, how about when the semicircle is given?
What is unit length?
Is it 1? Then, what is 1? Should it be 1cm? Or 1inch?
The unit length is a reference length. Thus, if we decided to make one segment be a reference, then we can say the length of the segment is 1.
Therefore, if the semicircle is given then the radius should be a unit length here.
By copying the semicircle, we can easily figure out the ratio 
.
Since the semicircle given, we fix 
=1, then 
. First, we have to construct 
, and then 
by using the property of similar triangles.
For the given circle below,
Construct a perpendicular line so that we have a triangle of the length of the base is 1 as a diameter of the given circle, and the height 2.
The hypotenuse is . Using the hypotenuse, we can construct .
By the property of similar triangles, the length of the base of the orange triangle is . If we can construct the right triangle of hypotenuse is the diameter of the given circle and ont side is , we can construct the inscribed square in a given semicircle.
By copying the right triangle, we can have the inscribe rectangle of the given circle of dimension
and
.
If we cut it in half, we will get the inscribed square in the given semicircle.
