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Let \( a \), \( b \) be positive real numbers. If \( a, A_{1}, A_{2}, b \) are in arithmetic progression, \( a, G_{1}, G_{2}, b \) are in geometric progression and \( \mathrm{a}, \mathrm{H}_{1}, \mathrm{H}_{2} \), b are in harmonic progression, show that
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\[
\frac{\mathrm{G}_{1} \mathrm{G}_{2}}{\mathrm{H}_{1} \mathrm{H}_{2}}=\frac{\mathrm{A}_{1}+\mathrm{A}_{2}}{\mathrm{H}_{1}+\mathrm{H}_{2}}=\frac{(2 \mathrm{a}+\mathrm{b})(\mathrm{a}+2 \mathrm{~b})}{9 \mathrm{ab}} \text {. }
\]
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