Find the orthocenter of the triangle whose vertices are $(0, 0), (2, -1),$ and $(1, 3).$

If $\left( \frac{3}{2}, 0 \right)$,$\left( \frac{3}{2}, 6 \right)$,and $(-1, 6)$ are the midpoints of the sides of a triangle,find the incenter of the triangle.

Let $A(1,1), B(1,-1), C(-1,1)$ be the vertices of $\triangle ABC$. Let $S$ be the circumcentre,$O$ be the orthocentre,and $I$ be the incentre of the $\triangle ABC$. Then $IS + OS =$ ?

Let the vertex of a triangle $ABC$ be $A(1, 2)$,and the mid-point of the side $AB$ be $M(5, -1)$. If the centroid of this triangle is $G(3, 4)$ and its circumcenter is $O(\alpha, \beta)$,then $21(\alpha + \beta)$ is equal to:

The incentre of a triangle with vertices $(7, 1)$,$(-1, 5)$,and $(3 + 2\sqrt{3}, 3 + 4\sqrt{3})$ is

If in triangle $ABC$,$A \equiv (1, 10)$,circumcentre $\equiv \left( -\frac{1}{3}, \frac{2}{3} \right)$ and orthocentre $\equiv \left( \frac{11}{3}, \frac{4}{3} \right)$,then the coordinates of the mid-point of the side opposite to $A$ are: