If one Arithmetic Mean (A.M) $A$ and the two Geometric Means (G.M.'s) $G_1$ and $G_2$ are inserted between two given positive numbers, prove that:

Question

If one Arithmetic Mean (A.M) $A$ and the two Geometric Means (G.M.'s) $G_1$ and $G_2$ are inserted between two given positive numbers, prove that: $$\dfrac {{G_1}^2}{G_2} + \dfrac {{G_2}^2}{G_1}=2A$$

My Attempt: Let $a$ and $b$ are any two positive numbers. Then, $$A=\dfrac {a+b}{2}$$ Also, $a, G_1, G_2, b$ are in G.P. so, $\dfrac {G_1}{a}=\dfrac {G_2}{G_1}=\dfrac {b}{G_2}$

How do I proceed further?

Answer 1

$$\implies G_1^2=aG_2,G_2^2=bG_1$$

$$\implies\dfrac{G_1^2}{G_2}+\dfrac{G_2^2}{G_1}=a+b=2A$$

Answer 2

let $$\frac{G_1}{a}=\frac{G_2}{G_1}=\frac{b}{G_2}=t$$ then $$G_1=at$$ $$G_2=at^2$$ $$b=t^3a$$ then we have the equation $$\frac{a^2t^2}{t^2a}+\frac{t^4 a^2}{at}=2A$$ can you finish? simplifying we get $$a+t^3a=a+b$$ and remember that $$b=t^3a$$