Let numbers be (\alpha,\beta).

Two A.Ms: (A_1=\dfrac{2\alpha+\beta}{3}), (A_2=\dfrac{\alpha+2\beta}{3})

[ A_1+A_2=\alpha+\beta ]

Two G.Ms: (G_1=\sqrt[3]{\alpha^2\beta}), (G_2=\sqrt[3]{\alpha\beta^2}), (G_1G_2=\alpha\beta)

Two H.Ms: reciprocals insert two A.Ms between (1/\alpha) and (1/\beta).

[ \dfrac{A_1+A_2}{G_1G_2}=\dfrac{\alpha+\beta}{\alpha\beta}=\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{H_1+H_2}{H_1H_2} ] (after verifying the H-side identity).

Proved.