Constructing A Square From A Rectangle

The
Constructing A Square From A Rectangle
Problem

Given: the rectangle in the figure above, construct a square
whose area equals the area of the given rectangle, Area = ab.

First construct the line segment AB, with length a + b as follows:

and then bisect the line segment at point O.

            Now draw a circle with radius equal to half the length of line segment just drawn, centered at point O,

            Circle And Chords

            and construct a perpendicular line segment to the segment AB at point P as shown.

            Because of properties of chords, angles formed by chords, and similar triangles, the triangles APC and DPB
            are similar and the sides form a proportion, such that, x is the geometric mean of a and b,
            or a is to x as x is to b, that is, a/x = x/b.
            In other words, x² = ab, the area of the given rectangle.

            Now draw a square with sides equal in length to x, it has the same area as the given rectangle.

            Square

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