plz explain!If o is the circumcentre & o' the ortocentre of a triangle ABC, prove that :1) SA+SA+SC = 3SG, where S is any point in the plane of ABC2) OA+OB +OC = OO'3)O'a+O'B+O'C = 2OO'4) AO'+O'B+O'C=APWHERE AP IS DIAMETER OF CIRCUMCIRCLE. ALL ARE IN VECTOR FORMS

DEAR neeraj pawar

according to definition of origin, it is any point in the space. so if a relation is valid with origin, it will be valid for every other point in the space.

the vector form of centroid is OG={OA+OB+OC}/3.

SINCE IT IS VALID FOR ORIGIN "O" {G IS THE CIRCUMCENTRE} IT IS VALID FOR ANY OTHER POINT .

THEREFORE SA+SB+SC = 3SG.

SIMILARLY YOU CAN DO THE OTHER SUMS BY JUST REPLACING ORIGIN BY ANY OTHER REQUIRED POINT {YOU CAN REPLACE ORIGIN BY EVEN THE POINTS A, B, C, O, O' AS IN YOUR QUESTION.}

THANK YOU..