In which ratio does the point $P (-4, 3)$ divide the line segment joining the points $A (1, -2)$ and $B (-6, 5)$?

The vertices of $\Delta ABC$ are $A(a, 6)$, $B(5, 1)$, and $C(4, 6)$. If the circumcentre of the triangle is $P(2, 3)$, find the value of $a$.

Show that the centroid of the triangle with vertices $(a, b-c)$,$(b, c-a)$,and $(c, a-b)$ lies on the $X$-axis.

If both the coordinates of the centroid of a triangle with vertices $(a, 2)$,$(2, -a)$,and $(0, 2)$ are equal,then $a=$...........

If $G$ is the centroid of a triangle having vertices $A(3h, 3k)$,$B(-3a, 0)$,and $C(3a, 0)$,then prove that $AB^2 + BC^2 + AC^2 = 3(GA^2 + GB^2 + GC^2)$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $MN = 3$ and $BC = 7$,then $M$ divides $\overline{AB}$ from $A$ in the ratio.......