Special Points in Triangles

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

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

If $\left(\alpha , \beta \right)$ is the orthocenter of the triangle $ABC$ with vertices $A\left(3, –7\right), B\left(–1, 2\right)$ and $C\left(4, 5\right)$, then $9\alpha -6\beta +60$ is equal to

Let $C(\alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4x+3y=69$, $4y-3x=17$, and $x+7y=61$. Then $(\alpha -\beta {)}^2+\alpha +\beta$ is equal to

The orthocentre of the triangle having vertices $A\left(1,2\right), B\left(3,-4\right)$ and $C\left(0,6\right)$ is

Let in triangle $ABC$, $A=45°, B=60°, C=75°$ then the ratio in which the orthocentre divides the altitude $AD$ is

The circumcentre of the triangle formed by the points $\left(a\cos \alpha ,a\sin \alpha \right), \left(a\cos \beta ,a\sin \beta \right) \& \left(a\cos \gamma ,a\sin \gamma \right)$ is 

A right triangle has sides '$a$' and '$b$' where $a>b$. If the right angle is bisected then find the distance between orthocentres of the smaller triangles using coordinate geometry.

Number of right isosceles triangles that can be formed with points lying on the curve $8x^3+y^3+6xy=1$ is

The incentre of the triangle with vertices $\left(1\text{,}\sqrt{3}\right)$ , $(0,0)$ and $(2,0)$ is

The orthocentre of the triangle with vertices $\left(6,-1\right),\mathrm{ }\left(-2,-1\right)$ and $\left(2,5\right)$ is

The orthocentre of the triangle with vertices $\left(-2,-6\right),\mathrm{ }\left(-2,4\right)$ and $\left(1,3\right)$ is

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

The point that is equidistant from the vertices of the triangle is called

If the orthocentre and the centroid of a triangle are $(-3,5,2)$ and $(3,3,4)$ respectively, then its circumcentre is

The equation of the line joining the centroid with the orthocentre of the triangle formed by the points $\left(-2, 3\right), \left(2,-1\right), \left(4,0\right)$ is

The triangle whose orthocentre, circumcentre and incentre coincide is known as

Gravitation centre of a triangle

In $△ABC$, medians $AD, BE$ and $CF$ intersect each other at$G$. Prove that $AD+BE>\frac{3}{2}AB$.