need a urgent and complete solution for this

**kamaksh_ice** · September 1st 2007, 06:56 PM

prove thatA.M, H.M
,AND G.M of two positive nos. 'a' and 'b' can be lengths of right angled triangle if and only a=b(G^3)
where G=0.5*(1+(5^0.5))

**red_dog** · September 1st 2007, 11:05 PM

We have $A.M.\geq G.M.\geq H.M.$
So A.M., G.M., H.M. can be lengths of a right angled triangle if and only if
$\displaystyle\left(\frac{a+b}{2}\right)^2=(\sqrt{a b})^2+\left(\frac{2ab}{a+b}\right)^2$
This is equivalent to $a^4-18a^2b^2+b^4=9$
Divide the equation by $b^4$ and note $\displaystyle\left(\frac{a}{b}\right)^2=t$
$t^2-18t+1=0\Rightarrow t_{1,2}=9\pm 4\sqrt{5}=(\sqrt{5}\pm 2)^2$
We can suppose that $a\geq b\Rightarrow\frac{a}{b}\geq 1$
So, $t=(2+\sqrt{5})^2$ .
$\displaystyle\left(\frac{a}{b}\right)^2=(2+\sqrt{5 })^2\Rightarrow\frac{a}{b}=2+\sqrt{5}\Rightarrow a=b(2+\sqrt{5})$

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