Special Points in a Triangle

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

A triangle has angles $90°,60°$ and $30°$. The orthocentre of the triangle will lie on the _____  of the triangle.

A triangle has angles $110°,30°$ and $40°$. The orthocentre of the triangle will lie _____ the triangle.

A triangle has angles $50°,60°$ and $80°$. The orthocentre of the triangle will lie _____ the triangle.

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 $A(5,2), B(10,12)$ and$P(x,y)$ are such that $\frac{AP}{PB}=\frac{3}{2}$, then the internal bisector of $\angle APB$ always passes through

Orthocentre of the triangle formed by the lines $x+y=1$ and $xy=0$ is

If $G\left(\overrightarrow{g}\right),H\left(\overrightarrow{h}\right)$ and $P\left(\overrightarrow{p}\right)$ are centroid, orthocenter and circumcenter of a triangle and $x\overrightarrow{p}+y\overrightarrow{h}+z\overrightarrow{g}=0$ then $\left(x, y, z\right)=$

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 $AD,BE$ and $CF$ are three medians of the $∆ABC$ then $AD+BE<CF$.

Let $BE$and $CF$ be the two medians of a $△ABC$ and $G$ be their intersection. Also let $EF$ cut $AG$ at $O$. Then $AO:OG$ is $3:1$.

$\mathrm{AD}$ is the median of the triangle $\mathrm{ABC}$ and $\mathrm{G}$ is the centroid of $△\mathrm{ABC}$. Then $\mathrm{AD}:\mathrm{AG}$ is $3:2$.

The locus of the centroid of the triangle whose vertices are $\left(3\cos \alpha ,3\sin \alpha \right)$, $\left(9\sin \alpha ,-9\cos \alpha \right)$ and $\left(1,0\right)$ is a circle of radius $R$, then the value of $\frac{9R^2}{2}$ is equal to

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 bisectors of angles $\mathrm{A}$ and $\mathrm{B}$ of a scalene triangle $\mathrm{ABC}$ meet at $\mathrm{O}.$ What is the point $\mathrm{O}$ called? Incentre$/$ Incircle$/$ Circumcircle.

Perpendicular bisectors of the sides $\mathrm{AB}$ and $\mathrm{AC}$ of a triangle $\mathrm{ABC}$ meet at $\mathrm{O}$. What do you call the point $\mathrm{O}$? Circumcenter $/$ Circumradius $/$ Circumcircle.

Point G is the centroid of $∆ ABC$.

 If $l\left(AP\right) =6$ then $l\left(GP\right) =$?