Arithmetic Mean, Geometric Mean, Harmonic Mean, Root Mean Square

The figure above shows a semicircle with diameter AB and center O. C is a point on AB, CE and OD are perpendicular  to AB, and CF is perpendicular to OE. If AC = a and CB = b, prove that:

• OD is the arithmetic mean of a and b; AM(a,b).

• CE is the geometric mean of a and b; GM(a,b).

• EF is the harmonic mean of a and b; HM(a,b).

• CD is the root mean square of a and b; RMS(a,b)

• Inequality: RMS (a,b) > AM (a,b) > GM (a,b) > HM (a,b)

• .