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

Q.

If $x_1,x_2,x_3$ as well as $y_1,y_2,y_3$ are in G.P. with the same common ratio, then the points $(x_1,y_1),(x_2,y_2)\mathrm{and}(x_3,y_3)$:

1. Lie on a straight line

2. Lie on a circle

3. Lie on an ellipse

4. Are vertices of a triangle

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,

1. $\frac{10\hat{i}-13\hat{j}}{3}$
2. $\frac{-10\hat{i}+13\hat{j}}{3}$
3. $\frac{10\hat{i}+13\hat{j}}{3}$
4. $\frac{-10\hat{i}-13\hat{j}}{3}$

Q. Given the vertices of triangle by position vectors $\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{k}and\hat{j}+\hat{k}$ the centroid and Incentre of the triangle will be given by

1. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+1)\hat{i}+(\sqrt{2}+1)\hat{j}+(\sqrt{2}+2)\hat{k}}{2+\sqrt{2}}$
2. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+1)\hat{i}+(\sqrt{2}+2)\hat{j}+(\sqrt{2}+1)\hat{k}}{2+\sqrt{2}}$
3. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+1)\hat{i}+(\sqrt{2}+2)\hat{j}+(\sqrt{2}+2)\hat{k}}{2+\sqrt{2}}$
4. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+2)\hat{i}+(\sqrt{2}+1)\hat{j}+(\sqrt{2}+2)\hat{k}}{2+\sqrt{2}}$

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

1. $I=\frac{a.\overline{p_1}+a.\overline{p_2}+c.\overline{p_3}}{a+b+c}$
2. $I=\frac{b.\overline{p_1}+c.\overline{p_2}+a.\overline{p_3}}{a+b+c}$
3. $I=\frac{c.\overline{p_1}+a.\overline{p_2}+b.\overline{p_3}}{a+b+c}$
4. $I=\frac{\overline{p_1}+\overline{p_2}+\overline{p_3}}{a+b+c}$

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,

1. $\frac{10\hat{i}-13\hat{j}}{3}$
2. $\frac{-10\hat{i}+13\hat{j}}{3}$
3. $\frac{10\hat{i}+13\hat{j}}{3}$
4. $\frac{-10\hat{i}-13\hat{j}}{3}$