Let $a$ and $b$ are positive real numbers. If $a, A_1, A_2, b$ are in arithmetic progression $a, G_1, G_2, b$ are in G.P. and $a, H_1, H_2, b$ are in H.P. show that $\frac{G_1 G_2}{H_1 H_2}=\frac{A_1+A_2}{H_1+H_2}=\frac{(2 a+b)(a+2 b)}{9 a b}$