If $(2, 3)$,$(4, 5)$,and $(-2, 11)$ are the vertices of a triangle,what is the distance between the vertex $(4, 5)$ and the circumcenter?

In $\Delta ABC$,the coordinates of $B$ are $(0, 0)$,$AB = 2$,$\angle ABC = \frac{\pi}{3}$,and the coordinates of the midpoint of $BC$ are $(2, 0)$. Find the centroid of the triangle.

Two vertices of a triangle are $(5, -1)$ and $(-2, 3)$. If the origin $(0, 0)$ is the orthocentre of this triangle,then the coordinates of the third vertex of that triangle are

Find the orthocenter of the triangle with vertices $(8, -2)$,$(2, -2)$,and $(8, 6)$.

The mid-point of the line segment joining the centroid and the orthocentre of the triangle whose vertices are $(a, b), (a, c)$ and $(d, c)$ is

$(-2, -1)$ and $(2, 5)$ are two vertices of a triangle and $\left(2, \frac{5}{3}\right)$ is its orthocenter. If $(m, n)$ is the third vertex of that triangle,then $m+n=$