If (a2+b2 +c2)(x2+y2 +z2) = (ax+by+cz)2 then prove that a(x+y+z) = x(a+b+c)

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Please find below the solution to the asked query:

We have:a2+b2+c2x2+y2+z2=ax+by+cz2a2x2+a2y2+a2z2+b2x2+b2y2+b2z2+c2x2+c2y2+c2z2=a2x2+b2y2+c2z2+2abxy+bcyz+cazxa2y2+a2z2+b2x2+b2z2+c2x2+c2y2=2abxy+2bcyz+2cazxa2y2+b2x2-2abxy+b2z2+c2y2-2bcyz+c2x2+a2z2-2cazx=0ay-bx2+bz-cy2+az-cx2=0Sum of squares of three quantities can be 0 if and only if all three are 0.ay-bx=0 and bz-cy=0 and az-cx=0ay=bx ;ibz=cy iiaz=cx ;iiiNowax+y+z=ax+ay+az=ax+bx+cx  Using i and iii=xa+b+cHence:ax+y+z=xa+b+c Proved

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