Centroid of a Triangle

The coordinates of the centroid of a triangle and those of two of its vertices are respectively $\left(\frac{2}{3},2\right),\left(2,3\right),\left(-1,2\right)$. Find the area of the triangle.

Find the centroid and incentre of the triangle whose vertices are $\left(2,4\right),\left(6,4\right),\left(2,0\right)$.

The area of a triangle is $\frac{3}{2}$ square units. Two of its vertices are the points $A(2,-3)$ and $B(3,-2);$ the centroid of the triangle lies on the line $3 x-y-8=0 .$ Find the third vertex $C$.

Two vertices of a triangle are $A\left(2,1\right)$ and $B\left(3,-2\right).$The third vertex $C$ lies on the line $y=x+9$. If the centroid of $\mathrm{\Delta }ABC$ lies on $y$-axis, find the coordinates of $C$ and the centroid.

If $P,Q,R$ divide the sides $BC,CA$ and $AB$ of $\Delta ABC$ in the same ratio, prove that the centroid of the triangles $ABC$ and $PQR$ coincide.

If $G$ be the centroid of a triangle $ABC$, prove that: $A{B}^2+B{C}^2+C{A}^2=3\left(G{A}^2+G{B}^2+G{C}^2\right)$

The area of a triangle is $3$ square units. Two of its vertices are $A(3,1), B(1,-3)$ and the centroid of the triangle lies on $x$-axis. Find the coordinates of the third vertex $C$.

Two vertices of a triangle are $(1,4)$ and $(5,2)$. If its centroid is $(0,-3)$, find the third vertex.

The coordinates of the centroid of a triangle are $(\sqrt{3},2)$, and two of its vertices are $(2\sqrt{3},-1)$ and $(2\sqrt{3},5)$. Find the third vertex of the triangle.

Find the centroid of the triangle $ABC$ whose vertices are $A (9,2), B(1,10)$ and $C(-7,-6)$. Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the two triangles have the same centroid?

If $(1,2), (0,-1)$ and $(2,-1)$ are the middle points of the sides of the triangle, find the coordinates of its centroid.