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

Online Geometry

Geometric, Arithmetic, Harmonic Mean

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)

  • Geometric, Arithmetic, Harmonic Mean Relationship

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