Summation of Vectors help!

Let the vertices of the triangle ABC be A (a) ; B (b) and C (c) w.r.t origin.

Let O ' (x,y) be the orthocentre and O (0,0) be the circumcentre (origin).
Let G be the centroid.

Thus G\ =\ \left(\frac{\vec{a}\ +\ \vec{b}\ +\ \vec{c} }{3} \right)

Since the centroide G divides the orthocentre C (x,y) and circumcentre (0,0) in the ratio 2 : 1 , thus by using section formula , we get

\left(\frac{\vec{a}\ +\ \vec{b}\ +\ \vec{c} }{3} \right)\ =\ \frac{2\ (0)\ +\ OO'}{3}

Thus, we get

\vec{a}\ +\ \vec{b}\ +\ \vec{c}\ =\ OO'

or

OA + OB + OC = OO'

Thus, (i) proved.

(ii) Now, consider triangle AOO' ,

using the rule of vectors addition , we get

O'A\ = O'O\ +\ OA

O'A\ = O'O\ +\ \vec{a} .......................(ii)

Similarily, we get

O'B\ = O'O\ +\ \vec{b} .....................(ii)

&

O'C\ = O'O\ +\ \vec{c} .....................(iii)

From (i), (ii) and (iii), we get

O'A\ +\ O'B\ + \ O'C\ = 3\ O'O \ +\ (\vec{a}\ +\ \vec{b}\ +\ \vec{c})

Since Since\ \vec{a}\ +\ \vec{b}\ +\ \vec{c}\ =\ OO'

Thus, we get

O'A + O'B + O'C = 2 OO'

Thus proved