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## Homework Statement

If [itex]\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b},[/itex] prove that [itex]x^{b+c-a}.y^{c+a-b}.z^{a+b-c}=1[/itex]

## Homework Equations

## The Attempt at a Solution

I solved a question similar to it in a way. So I tried this in the same way. The only difference between the two questions is that except a,b and c, the denominators are also x, y and z.

I didn't know if it would work.

So I tried like this.

Let [itex] m= b-c, n= c-a [/itex]then [itex] a-b=-(m+n)[/itex]

By cross multiplication, I got

[itex]z^m=x^{-(m+n)}\\z^n=y^{-(m+n)}\\x^n=y^m\\\\\Rightarrow z=1/xy, x=1/yz, y=1/xz[/itex]

In my old question it was to prove [itex]x^x.y^y.z^z=1[/itex]. So I substituted easily.

I am struggling to substitute here.

So I tried writing [itex]x^{b+c-a}.y^{c+a-b}.z^{a+b-c}[/itex] it easier by using m and n.

[itex]x^{b+c-a}.y^{c+a-b}.z^{a+b-c}=x^{2n+m+a}.y^{a-m}.z^{a+m}[/itex]

I am struggling then after. Can anyone give me a hint.

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