Let O be the point inside triangle △ABC, and A1, B1, C1 are middle of the sides AB, BC, CD.
How can I prove that (using vectors):
OA1 + OB1 + OC1 = OA + OB + OC
2026-04-03 03:16:00.1775186160
Show that OA1 + OB1 + OC1 = OA + OB + OC using vectors
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Let $\vec a, \vec b, \vec c$ be the position vectors of A,B,C about O then PVs of A1,B1,C1 are $(\vec b +\vec c)/2, (\vec a+ \vec c)/2, (\vec a+ \vec b)/2,$ respectively. Then LHS is $$LHS= (\vec b +\vec c)/2+(\vec a+ \vec c)/2+(\vec a+ \vec b)/2= \vec a+ \vec b +\vec c= RHS$$