Given,

Position vector of circumcentre, $$\overrightarrow C = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}$$

We know, position vector of centroid, $$\overrightarrow G = {{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 3}$$

Now, let $$\overrightarrow R $$ be the orthocentre of the triangle.

We know, $$\overrightarrow G $$ $$ = {{2\overrightarrow C + \overrightarrow R } \over 3}$$

$$ \Rightarrow $$   3$$\overrightarrow G $$ $$ = 2\overrightarrow C + \overrightarrow R $$

$$ \Rightarrow $$   $$\overrightarrow R = 3\overrightarrow G - 2\overrightarrow C $$

=   $$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right) - 2\left( {{{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}} \right)$$

=   $${{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 2}$$