The orthocentre of the triangle formed by the lines $x + y = 1$,$2x + 3y = 6$,and $4x - y + 4 = 0$ lies in which quadrant?

If two vertices of a triangle are $(5, -1)$ and $(-2, 3)$ and its orthocentre is at $(0, 0)$,then the third vertex is:

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 $\frac{(c_{1}-c_{2})^{2}}{2|m_{1}-m_{2}|}$.

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:

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

The mid-point of the line segment joining the centroid and the orthocentre of the triangle whose vertices are $(a, b), (a, c)$ and $(d, c)$ is