if G is the centroid of triangle ABC and a,b,c are side lenghts then prove that a2+b2c2=3(OA2+OB2+OC2)- 9 OG2 where O is any point in plane of triangle ABC....kindly give proof using vector algebra rather than plane geometry (i know all concepts related to vector analysis and vector algebra)

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Please find below the solution to the asked query:

Let O be origin, so we havePosition vectors of A,B,C are a,b,c, henceOA=aOB=bOC=cHenceOG=OA+OB+OC3=a+b+c33OA2+OB2+OC2-9OG2=3a2+b2+c2-9a+b+c32=3a2+b2+c2-a+b+c23OA2+OB2+OC2-9OG2=3a2+b2+c2-a2+b2+c2+2a.b+b.c+c.a...iWe know that for a trianglea+b+c=0Take dot product with a on both sidesa2+a.b+c.a=0Similarlyb2+a.b+b.c=0c2+b.c+c.a=0Add three equationa2+b2+c2=-2a.b+b.c+cPut in i3OA2+OB2+OC2-9OG2=3a2+b2+c2-a2+b2+c2+2a.b+b.c+c.a=3a2+b2+c2-a2+b2+c2+a2+b2+c23OA2+OB2+OC2-9OG2=a2+b2+c2

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