Let ABC be a triangle whose circumcentre is at P. If the position vectors of A,B,C and P are $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{4}$ respectively then the position vector of the orthocentre of the triangle is 

Circumcentre $\overrightarrow{OP}=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{4}$

And we know that 

Centroid, $\overrightarrow{OG}=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{3}$

We have orthocentre, centroid and circumcentre are collinear and G divides the line in 2:1 ratio

$\overrightarrow{OG}=\frac{2\overrightarrow{OP}+\overrightarrow{\text{OH}}}{3}$

$\overrightarrow{OH}=3\overrightarrow{OG}-2\overrightarrow{OP}$

$O\overrightarrow{H}=\left(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right)-\left(\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{2}\right)=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{2}$