The circumcentre of the triangle formed by the points $A(1, \sqrt{3})$,$B(-1, -\sqrt{3})$,and $C(3, -\sqrt{3})$ is

If $A(4, -3)$,$B(3, -2)$,and $C(2, 8)$ are the vertices of a triangle,then its centroid will be

If the vertices $P, Q, R$ of a triangle $PQR$ are rational points (points with rational coordinates),which of the following points is/are not necessarily a rational point?

Let the vertex of a triangle $ABC$ be $A(1, 2)$,and the mid-point of the side $AB$ be $M(5, -1)$. If the centroid of this triangle is $G(3, 4)$ and its circumcenter is $O(\alpha, \beta)$,then $21(\alpha + \beta)$ is equal to:

Show that the area of the triangle formed by the lines $y=m_{1}x+c_{1}$,$y=m_{2}x+c_{2}$ and $x=0$ is $\frac{(c_{1}-c_{2})^{2}}{2|m_{1}-m_{2}|}$.

If the vertices of a triangle are $(a, 1), (b, 3),$ and $(4, c),$ then the centroid of the triangle will lie on the $x$-axis if