If the vertices of a triangle are $(1, 2, 3)$,$(2, 3, 1)$,and $(3, 1, 2)$,and if $H, G, S$,and $I$ respectively denote its orthocenter,centroid,circumcenter,and incenter,then $H+G+S+I$ is equal to:

What is the angle between two diagonals of a cube?

$\Pi_1, \Pi_2, \Pi_3$ are three planes which are respectively parallel to the $YZ, ZX$ and $XY$ planes at distances $a, b$ and $c$ forming a rectangular parallelopiped. $d_1$ is a diagonal of the face of $XY$-plane not passing through the origin and $d_2$ is a diagonal of the plane $\Pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelopiped are negative,then the angle between $d_1$ and $d_2$ is

The angle between two diagonals of a cube is

If $A=(2,3,4)$ and $B=(-2,3,4)$,then the locus of a point $P(x,y,z)$ such that $PA+PB=4$ is

Let $ABCD$ be a tetrahedron such that the edges $AB, AC$ and $AD$ are mutually perpendicular. Let the areas of the triangles $ABC, ACD$ and $ADB$ be $5, 6$ and $7$ square units respectively. Then the area (in square units) of the $\triangle BCD$ is equal to :