Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

Q. Given position vectors $\overline{a},\overline{b},\overline{c}$ of points A, B, C for a triangle. The centroid can be given by

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Q. Given orthocentre $\overline{H}$ and circumcentre $\overline{C}$ for a triangle as $2\hat{i}+3\hat{j}and4\hat{i}+5\hat{k}$.Then the centroid of triangle can be given by,

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Q. Given 3 points with position vectors $\overline{p_1},\overline{p_2}and\overline{p_3}$ which form the vertices of a triangle with side lengths $a=|\overline{p_2}-\overline{p_1}|,b=|\overline{p_3}-\overline{p_2}|,c=|\overline{p_1}-\overline{p_3}|$. Then the in-centre is given by

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