Relation between O,G,C

Q.

If a triangle has its orthocenter at (1, 1) and circumcenter at $(\frac{3}{2},\frac{3}{4})$ , then the coordinates of the centroid of the triangle are

1. $(\frac{4}{3},-\frac{5}{6})$

2. $(\frac{4}{3},\frac{5}{6})$

3. $(-\frac{4}{3},\frac{5}{6})$

4. $(-\frac{4}{3},-\frac{5}{6})$

Q. If $A(7,9),B(3,-7)$ and $C(-3,3)$ are the vertices of a triangle and $S(\alpha ,\beta ),O(l,m)$ are its circumcentre and orthocentre respectively, then the square of the distance between $P(m,\alpha )$ and $Q(\beta ,l)$ is

Q. If $A(\cos \alpha ,\sin \alpha ),$ $B(\cos \beta ,\sin \beta ),$ $C(\cos \gamma ,\sin \gamma )$ are the vertices of $△ABC$ and $H,G,S$ are the orthocentre, centroid and circumcentre of $△ABC$ respectively, then which of the following is/are true?

1. $S\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
2. $G\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
3. $H\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
4. $S\equiv (0,0)$

Q. If ($\alpha ,\beta$), $(\overline{x},\overline{y})and(u,v)$are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle.

1. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta +v$
2. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta -v$
3. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta +v$
4. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta -v$