Means as chords

 

Given two positive numbers a and b, there exists four very known means, namely the harmonic, geometric, arithmetic and quadratic mean of a and b, given respectively by the formulas:

h=\frac{2}{\frac{1}{a}+\frac{1}{b}}, \quad g=\sqrt{ab}, \quad m=\frac{a+b}{2}, \quad q=\sqrt{\frac{a^2+b^2}{2}}.

It is also well known that we always have

The following figure shows two lengths a and b, together with the four means h, g, m, q, as chords CA, CB, CH, CG, CM, CQ respectively of a same circle.

Remarks:

  • We start with the segment CB = b.
  • Let N be the midpoint of CB. Erect a perpendicular NO = a/2.
  • The circle with center N and radius NO meet the segment NB at E.
  • Draw the circle G with center O through B and C.
  • Draw the circle with center C and radius a. This circle meet the segment CB at D, and the circle (O) at A.
  • The internal bisector of angle ACB meets the circle with center O at Q.
  • We find the point M on (O) such that CE=CM.
  • The perpendicular though AB to D meet CM at J, and G at G.
  • The circle with center C and radius CJ meet G at H.
  • We also get that DG, CQ and AB are concurrent.
In the following figure, where the chord AB is parallel to the radius OC, we can see that the chord CQ is the quadratic mean of the chords CA and CB. The notation P(q) means that P is the point on the circle (O) such that OP makes an angle q with the horizontal.