x+y+z=0
=>x+y=-z; x+z=-y; y+z=-x
\(x^2+y^2-z^2=\left(x+y\right)^2-2xy-z^2\)
\(=\left(-z\right)^2-2xy-z^2=-2xy\)
\(x^2+z^2-y^2\)
\(=\left(x+z\right)^2-2xz-y^2\)
\(=\left(-y\right)^2-2xz-y^2=-2xz\)
\(y^2+z^2-x^2\)
\(=\left(y+z\right)^2-2yz-x^2\)
\(=\left(-x\right)^2-x^2-2yz=-2yz\)
\(P=\frac{xy}{x^2+y^2-z^2}+\frac{xz}{x^2+z^2-y^2}+\frac{yz}{y^2+z^2-x^2}\)
\(=\frac{xy}{-2xy}+\frac{xz}{-2xz}+\frac{yz}{-2yz}=-\frac12-\frac12-\frac12=-\frac32\)
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