The orthocenter of an equilateral triangle is $(3, -2)$. If one of its sides lies on the $x$-axis,find the vertex of the triangle that does not lie on the $x$-axis.

The orthocentre of the triangle with vertices $\left( 2, \frac{\sqrt{3} - 1}{2} \right)$,$\left( \frac{1}{2}, -\frac{1}{2} \right)$,and $\left( 2, -\frac{1}{2} \right)$ is:

In a $\triangle ABC$,if the medians $AD$ and $BE$ are such that $AD=4$,$\angle DAB=\frac{\pi}{6}$ and $\angle ABE=\frac{\pi}{3}$,then the area of $\triangle ABC$ (in square units) is

$A$ tower stands vertically inside an acute-angled triangular park $\Delta PQR$. If the angle of elevation of the top of the tower from each corner of the park is the same,then in $\Delta PQR$,the foot of the tower is at the

$A$ triangle has a vertex at $(1, 2)$ and the midpoints of the two sides through it are $(-1, 1)$ and $(2, 3)$. Then the centroid of this triangle is

The circumcentre of a triangle lies at the origin and its centroid is the midpoint of the line segment joining the points $(a^2 + 1, a^2 + 1)$ and $(2a, -2a)$,where $a \ne 0$. Then for any $a$,the orthocentre of this triangle lies on the line: