If a $\triangle ABC$ has vertices $A(-1, 7)$,$B(-7, 1)$ and $C(5, -5)$,then its orthocentre has coordinates

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

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

Let $ABC$ be an equilateral triangle with orthocenter at the origin and the side $BC$ on the line $x+2\sqrt{2}y=4$. If the coordinates of the vertex $A$ are $(\alpha, \beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2}\beta|$ is

Let $PQR$ be an acute-angled triangle in which $PQ < QR$. From the vertex $Q$,draw the altitude $QQ_1$,the angle bisector $QQ_2$,and the median $QQ_3$,with $Q_1, Q_2, Q_3$ lying on $PR$. Then,

The distance (in units) between the circumcentre and the centroid of the triangle formed by the vertices $(1, 2)$,$(3, -1)$ and $(4, 0)$ is