If the vertices of a triangle are $(a, b - c)$,$(b, c - a)$,and $(c, a - b)$,then the centroid of the triangle lies

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

If the vertices $P, Q, R$ of a triangle $PQR$ are rational points,which of the following points of the triangle $PQR$ is (are) always a rational point $(s)$? ($A$ rational point is a point both of whose coordinates are rational numbers.)

Let $PS$ be the median of the triangle with vertices $P(2,2)$,$Q(6,-1)$,and $R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to $PS$ is:

The centroid of a triangle is $(2, 3)$ and two of its vertices are $(5, 6)$ and $(-1, 4)$. Find the third vertex of the triangle.

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.