If x = AD and y = BD, what segment = arithmetic mean of x and y? What segment = geometric mean? Which one is bigger? Are they ever equal? Drag point D and watch the measurements.
In this figure, the sum of x and y is the diameter AB, so the arithmetic mean (1/2)(x+y) equals the radius of the circle, either EA or EB or EF.
On the other hand, CD = sqrt (xy). This can be shown either from similar triangles in a right triangle (the angle ACB is a right angle, since it is inscribed in a semicircle) or from the theorem about products of segment lengths of intersecting chords (AD * DB = DC * DC).
Since the segment CD is always less than a radius except when C = E (in this case x= y), the geometric mean is always less or equal to the arithmetic mean, with equality only when x = y.
Example: Suppose you have money in a savings account compounding continuously. If you have $100 at the start of the year and $106 at the end of the year, the amount you had after six months was the geometric mean of $100 and $106, which is less than the arithmetic mean of $103. (The geometric mean = $100 * sqrt(1.06) = $102.956 approximately, so the difference is not great in this case. Where in the figure, approximately, would you drag point D so that the ratio y/x = 1.06?)
J. King, U. of Washington, 10/29/2004