If $(a, b - c)$,$(b, c - a)$,and $(c, a - b)$ are the vertices of a triangle,then where does the centroid of the triangle lie?

Let $O(0,0), P(3,4), Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR, PQR, OQR$ are of equal area. The coordinates of $R$ are

The equation of the line joining the centroid with the orthocentre of the triangle formed by the points $(-2, 3), (2, -1), (4, 0)$ is

Two vertices of a triangle are $(5, -1)$ and $(-2, 3)$. If the orthocentre is the origin,then the coordinates of the third vertex are:

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

In $\triangle ABC$,if $A=(1,2)$ and the equations of the medians through $B$ and $C$ are $x+y=5$ and $x=4$ respectively,then the area of $\triangle ABC$ is