ax+by+cz=0 -> (ax+by+cz)^2=0 -> a^2x^2+b^2y^2+c^2z^2=-2(axby+bycz+axcz) (1)
Ta có:
[tex]bc.(y-z)^2+ac.(x-z)^2+ab.(x-y)^2= bc( y^2 - 2yz + z^2) + ac( x^2 - 2xz + z^2) + ab( x^2 - 2xy + y^2)= bcy^2 + bcz^2 - 2bcyz + acx^2 + acz^2 - 2acxz + abx^2 + aby^2 - 2abxy= (bcy^2 + bcz^2 + acx^2 + acz^2 + abx^2 + aby^2 ) - (2abxy + 2aczx + 2bcyz)= bcy^2 + bcz^2 + acx^2 + acz^2 + abx^2 + aby^2 + a^2x^2 + b^2y^2 + c^2z^2[/tex]
(Theo (1)
[tex]= x^2( ac + ab + a^2) + y^2(bc + ab + b^2) + z^2(bc + ac + c^2)\\= ax^2( a + b + c ) + by^2( a + b + c ) + cz^2( a + b + c )\\= ( a + b + c )( a.x^2+b.y^2+c.z^2)[/tex]
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