which can be visualized in a semi-circle whose diameter is [AB] and center D.
Suppose AC = x1 and BC = x2. Construct perpendiculars to [AB] at D and C respectively. Join [CE] and [DF] and further construct a perpendicular [CG] to [DF] at G. Then the length of GF can be calculated to be the harmonic mean, CF to be the geometric mean, DE to be the arithmetic mean, and CE to be the quadratic mean. The inequalities then follow easily by the Pythagorean theorem.
Comparison of harmonic, geometric, arithmetic, quadratic and other mean values of two positive real numbers and
Tests
To infer the correct order, the four expressions can be evaluated with two positive numbers.
^Djukić, Dušan (2011). The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959-2009. Problem books in mathematics. International mathematical olympiad. New York: Springer. p. 7. ISBN978-1-4419-9854-5.