Ancient Greek means

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The ancient Greeks defined ten means in terms of the following proportions (see here):

Let a > m > b > 0. Then m represents

(1) the arithmetic mean of a and b if

\frac{a-m}{m-b}=\frac{a}{a};

(2) the geometric mean of a and b if

\frac{a-m}{m-b}=\frac{a}{m}=\frac{m}{b};

(3) the harmonic mean of a and b if

\frac{a-m}{m-b}=\frac{a}{b};

(4) the contraharmonic mean of a and b if

\frac{a-m}{m-b}=\frac{b}{a};

(5) the first contrageometric mean of a and b if

\frac{a-m}{m-b}=\frac{b}{m};

(6) the second contrageometric mean of a and b if

\frac{a-m}{m-b}=\frac{m}{a};

(7) the semicontraharmonic mean of a and b if

\frac{a-m}{a-b}=\frac{b}{a};

(8) the semicontrageometric mean of a and b if

\frac{a-m}{a-b}=\frac{m}{a};

(9) the semiharmonic mean of a and b if

\frac{a-b}{m-b}=\frac{a}{b};

(10) the semigeometric mean of a and b if

\frac{a-b}{m-b}=\frac{m}{b}.

The names for the last four means are mine.

The algebraic expressions for the ten means Mn(a, b) are

(1) the arithmetic mean M1(a, b)

\frac{a+b}{2};

(2) the geometric mean M2(a, b)

(ab)^{1/2};

Note that

M_2=(M_1M_3)^{1/2}

(3) the harmonic mean M3(a, b)

\frac{a \boxplus b}{2^{-1}}=\left ( \frac{a^{-1}+b^{-1}}{2} \right )^{-1}= \frac{2ab}{a+b}=\frac{(a+b)^2-(a^2+b^2)}{a+b};

(4) the contraharmonic mean M4(a, b)

\frac{a^2+b^2}{a+b}=\frac{(a+b)^2-2ab}{a+b};

(5) the first contrageometric mean M5(a, b)

\frac{a-b+\sqrt{(a-b)^2+4b^2}}{2};

(6) the second contrageometric mean M6(a, b)

\frac{b-a+\sqrt{(b-a)^2+4a^2}}{2};

(7) the semicontraharmonic mean M7(a, b)

\frac{a^2-ab+b^2}{a}=\frac{ab+(a-b)^2}{a};

(8) the semicontrageometric mean M8(a, b)

\frac{a^2}{2a-b}=\frac{a}{2-b/a};

(9) the semiharmonic mean M9(a, b)

\frac{2ab-b^2}{a}=\frac{a^2-(a-b)^2}{a};

(10) the semigeometric mean M10(a, b)

\frac{b+\sqrt{4ab-3b^2}}{2}.

Only the first four means are symmetric. All means are reflexive.