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Template:QM AM GM HM inequality visual proof.svg

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Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b[note 1]
  1. ^ If NM = a and PM = b. AM = AM of a and b, and radius r = AQ = AG.
    Using Pythagoras' theorem, QM² = AQ² + AM² ∴ QM = √AQ² + AM² = QM.
    Using Pythagoras' theorem, AM² = AG² + GM² ∴ GM = √AM² − AG² = GM.
    Using similar triangles, ⁠HM/GM⁠ = ⁠GM/AM⁠ ∴ HM = ⁠GM²/AM⁠ = HM.
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