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Dat (x-y)2+(y-z)2+(x-z)2=A
=(x+y)3+z3-3x2y-3xy2-3xyz / A
=(x+y+z).(x2+2xy+y2-xy-yz+z2)-3xy(x+y+z) / A
=(x+y+z).(x2+y2+z2-xy-yz-xz) /A
=2(x+y+z).(x2+y2+z2-xy-yz-xz) /2A
=(x+y+z)[ (x2-2xy+y2)+(y2-2yz+z2)+(x2-2xz+z2) / 2A
=(x+y+z).[ (x-y}2+(y-z)2+(x-z)2 ] /2A
=(x+y+z). A /2A
=x+y+z /2
1: \(=\dfrac{\left(x^2+2xy+y^2\right)-1}{\left(x^2+2x+1\right)-y^2}\)
\(=\dfrac{\left(x+y+1\right)\left(x+y-1\right)}{\left(x+1-y\right)\left(x+1+y\right)}=\dfrac{x+y-1}{x-y+1}\)
2: \(=\dfrac{\left(x^2-y^2\right)\left(x^2+y^2\right)}{\left(x+y\right)\left(x^2-xy+y^2\right)}=\dfrac{\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)}{\left(x+y\right)\left(x^2-xy+y^2\right)}\)
\(=\dfrac{\left(x-y\right)\left(x^2+y^2\right)}{x^2-xy+y^2}\)
3: \(=\dfrac{\left(x+y\right)^3+z^3-3xy\left(x+y\right)-3xyz}{2x^2+2y^2+2z^2-2xy-2yz-2xz}\)
\(=\dfrac{\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)}{2\left(x^2+y^2+z^2-xy-yz-xz\right)}\)
\(=\dfrac{x+y+z}{2}\)