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

$(-2, -1)$ and $(2, 5)$ are two vertices of a triangle and $\left(2, \frac{5}{3}\right)$ is its orthocenter. If $(m, n)$ is the third vertex of that triangle,then $m+n=$

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

If the midpoints of the sides of a triangle are $(0, 1)$,$(1, 1)$,and $(1, 0)$,find the $x$-coordinate of its incentre.

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:

Consider a triangle having vertices $A(-2, 3)$,$B(1, 9)$,and $C(3, 8)$. If a line $L$ passing through the circum-center of triangle $ABC$ bisects line $BC$ and intersects the $y$-axis at point $\left(0, \frac{\alpha}{2}\right)$,then the value of the real number $\alpha$ is $.....$