If G be the centroid of a triangle ABC and P be any other point in the plane, prove that
PA2 + PB2+ PC2 = GA2+ GB2+GC2
Let A , B and C be the vertices of triangle ABC with coordinates (x1,y1) (x2,y2),(x3,y3) respectively.
P be any general point with coordinates (x,y).Coordinates of Centroid will be ( (x1+x2+x3)/3 ,(y1+y2+y3)/3)
Apply distance formula to get PA2 PB2PC2 GA2 GB2GC2 and GP2
PA2 = (x-x1)2 + (y-y1)2
PB2 = (x-x2)2 + (y-y2)2
PC2 = (x-x3)2 + (y-y3)2
Similarly find GA2 GB2GC2 and GP2 substitute the values and simplify.
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