Centroid of a Triangle

Let $(\alpha ,\beta )$ be the centroid of the triangle formed by the lines $15x-y=82, 6x-5y=-4$ and $9x+4y=17$. Then $\alpha +2\beta$ and $2\alpha -\beta$ are the roots of the equation

The centroid of the triangle formed by the lines $x-3y+3=0,x+3y+3=0,x+y-1=0$ is

Centroid of a triangle divides its median in the ratio

The centroid of the triangle whose vertices are $(0, 0), (4, 0) \mathrm{and} (0, 3)$.

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

In $∆ABC,$ medians $\overline{AD}$ and $\overline{CE}$ intersect at $P, PE=1.5, PD=2$ and $DE=2.5$. What is the area of $2\left[AEDC\right]$?

If $G$ is the centroid of a $△ABC$ and $P$ is any other point in the plane, then $P{A}^2+P{B}^2+P{C}^2$ is equal to

Let $ABC$ be an acute angled triangle and $AD,BE$ and $CF$ are its median, where $E$ and $F$ are the points $\left(3,4\right)$ and $\left(1,2\right)$ respectively and centroid of  $∆ABC$ is $G\left(3,2\right),$ then the equation of the side $AB$ is

Let $ABC$ be an acute angled triangle and $AD,BE$ and $CF$ are its median, where $E$ and $F$ are the points $\left(3,4\right)$ and $\left(1,2\right)$ respectively and centroid of  $∆ABC$ is $G\left(3,2\right),$ then the co-ordinates of $D$ are

If the vertices of a triangle are given by $A\left(\alpha , \frac{1}{\alpha }\right), B\left(\beta , \frac{1}{\beta }\right)$ and $C\left(\gamma , \frac{1}{ \gamma }\right)$; where $\alpha , \beta , \gamma$ are the roots of the cubic equation  $x^3-7x^2+11x-3=0.$ Then the coordinates of the centroid of the triangle $ABC$ is _____

If the centroid and circumcenter of a triangle are $\left(3,3\right)$, $\left(6,2\right)$, respectively, then the orthocentre of the triangle is _____.

Which of the following options is only correct combination?

Equation of the locus of the centroid of the triangle whose vertices are $\left(a\cos k, a\sin k\right), \left(b\sin k,-b\cos k\right)$ and $\left(1, 0\right),$ where $k$ is a parameter, is

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

If the coordinates of the centroid of a triangle are$(1, 3)$ and two of its vertices are$(-7, 6)$ and $(8, 5)$, then the third vertex of the triangle is

The vertices of a triangle are $\left(6,\mathrm{ }0\right),\mathrm{ }\left(0,\mathrm{ }6\right)$ and $\left(6,\mathrm{ }6\right)$ , the distance between its circumcentre and centroid is

The mid-points of sides $AB,BC,CA$ of a $\mathrm{\Delta }ABC$ are $\left(6,-1\right),\mathrm{ }\left(-\mathrm{ }4,-3\right),\mathrm{ }\left(2,-\mathrm{ }5\right)$ respectively, then centroid of $\mathrm{\Delta }ABC$ is

If the middle points of the sides of a triangle be $\left(-2,\mathrm{ }3\right),\mathrm{ }\left(4,\mathrm{ }-3\right)$ and $\left(4,\mathrm{ }5\right)$ , then the centroid of the triangle is -

If the vertices of a triangle be $\left(\mathrm{a},\mathrm{ }1\right),\mathrm{ }\left(\mathrm{b},\mathrm{ }3\right)$ and $\left(4,\mathrm{ }\mathrm{c}\right)$ , then the centroid of the triangle will lie on $\mathrm{x}$ -axis, if -

Two vertices of a triangle are $\left(5,\mathrm{ }4\right)$ and $\left(-2,\mathrm{ }4\right)$ . If its centroid is $\left(5,\mathrm{ }6\right)$ then the third vertex has the coordinates -