Special Points of 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

Find the excenter opposite to vertex $B$ of triangle with vertices $A(20,15), B(0,0) \&\hspace{0.17em}C(-36,15)$.

Find the excenter opposite to vertex $A$ of triangle with vertices $A(20,15), B(0,0) \&\hspace{0.17em}C(-36,15)$.

$△OAB$ is an equilateral triangle with vertices $O\left(0,0\right)$, $A\left(3,0\right)$ and $B\left(\frac{3}{2},\frac{3\sqrt{3}}{2}\right)$. Find co-ordinates of incenter of $△OAB$.

$△OAB$ is an equilateral triangle with vertices $O\left(0,0\right)$, $A\left(2,0\right)$ and $B\left(1,\sqrt{3}\right)$. Find co-ordinates of incenter of $△OAB$.

$△OAB$ is an equilateral triangle with vertices $O\left(0,0\right)$, $A\left(1,0\right)$ and $B\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. Find co-ordinates of incenter of $△OAB$.

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

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.

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

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

In an isosceles $△ABC, AB=AC$ and $D$ is midpoint of $BC$. Prove that circumcentre, in centre, orthocentre and centroid all are collinear.

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

Let $ABC$ be a variable triangle such that $A(1, 2)$, $\mathrm{B}$ and $\mathrm{C}$ lie on the line $\mathrm{y}=\mathrm{x}+\lambda$ (where $\lambda$ is a variable). The locus of the orthocentre of triangle $ABC$ is a straight line. Find $y$-intercept of that straight line.

If $\mathrm{A}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right),\mathrm{B}\left({\mathrm{x}}_2,{\mathrm{y}}_2\right)$ and $\mathrm{C}\left({\mathrm{x}}_3,{\mathrm{y}}_3\right)$ are vertices of a triangle such that ${\mathrm{x}}_1^2+{\mathrm{y}}_1^2={\mathrm{x}}_2^2+{\mathrm{y}}_2^2={\mathrm{x}}_3^2+{\mathrm{y}}_3^2,$ then

If the sides of a triangle be along the lines $a_{1}x+b_{1}y+c_1=0, a_{2}x+b_{2}y+c_2=0$ and $a_{3}x+b_{3}y+c_3=0$ where
$a_i, b_i, c_i\in Q,$ then which of the following point(s) for given triangle must have necessarily rational coordinates?

Let the straight lines, $5x-3y+15=0$ and $5x+3y-15=0$ form a triangle with the $\text{X}-$ axis. Then the radius of the circle circumscribing this triangle is: