What is the distance from the origin to the centroid of the triangle formed by the points $(1, 1)$,$(0, -7)$,and $(-4, 0)$?

In a triangle $PQR$,the coordinates of the points $P$ and $Q$ are $(-2, 4)$ and $(4, -2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x - y + 2 = 0$,then the centre of the circumcircle of the $\Delta PQR$ is

If the orthocentre of the triangle,whose vertices are $(1,2), (2,3)$ and $(3,1)$ is $(\alpha, \beta)$,then the quadratic equation whose roots are $\alpha+4\beta$ and $4\alpha+\beta$ is:

Find the incenter of the triangle with vertices $(1, \sqrt{3})$,$(0, 0)$,and $(2, 0)$.

In a triangle $ABC$,the coordinates of $A$ are $(1, 2)$ and the equations of the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$ respectively. Then the area of $\Delta ABC$ (in sq. units) is

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