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Let $a,b,c,x,y,z\in\mathbb{R}$. Prove that $ \left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2\geq(c-a)^2+(c-b)^2$

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    I found this unsolved in http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=293472&p=1586692#p1586692 .2010-11-27

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With help from Maple, I got $\left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2-(c-a)^2-(c-b)^2$ equal to $\frac{(c(x^3+y^3+z^3)+(a-c)(x^2y+y^2z+z^2x)+(b-c)(x^2z+y^2x+z^2y)-3(a+b-c)xyz)^2}{(x-y)^2(y-z)^2(x-z)^2}$ which of course is $\ge 0$.

But with no help from a computer algebra, how would one prove this?