Uses
A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. If we make a semicircle with a diameter of a+b, then the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean, which can be proved using the Pythagorean theorem. This can be used to accomplish quadrature of a rectangle (since a square whose sides are equal to the geometric mean of the sides of a rectangle has the same area as the rectangle), and thus of any figure for which we can construct a rectangle of equal area, such as any polygon (but not a circle).
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