The orthocentre of the triangle formed by the lines $xy = 0$ and $x + y = 1$ is

The incentre of the triangle with vertices $(1, \sqrt{3}), (0, 0)$ and $(2, 0)$ is:

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

The medians $AD$ and $BE$ of a triangle with vertices $A(0, b)$,$B(0, 0)$,and $C(a, 0)$ are perpendicular to each other,if

The orthocentre of the triangle formed by the vertices $(0, 0)$,$(8, 0)$,and $(4, 6)$ is

If $O, G, S$ are respectively the orthocentre,centroid and circumcentre of a triangle whose vertices are $A(2,3), B(2,4)$ and $C(4,3)$,then $AO^2 + 9BG^2 + 4CS^2 =$