Use (log x)/(y-z)=(log y)/(z-x)=(log z)/(x-y) =k
Then, (log x)=(y-z)*k... eq 1
(log y)=(z-x)*k.... eq 2
(log z)=(x-y)*k.... eq 3
Now consider log(x^x*y^y*z^z)
log(x^x*y^y*z^z) = xlog(x) + ylog(y) + zlog(z) =
=x(y-z)*k + y( z-x)*k + z(x-y)*k from eqn 1,2,3
=0
I.e., log(x^x*y^y*z^z) =0 =log (1)
I.e., (x^x*y^y*z^z) = 1.... hence proved.