If $A(0,0,0), B(3,4,0), C(0,12,5)$ are the vertices of a triangle $ABC$,then the $x$-coordinate of its incentre is

The coordinates of the foot of the perpendicular drawn from point $P(1, 0, 3)$ to the line joining points $A(4, 7, 1)$ and $B(3, 5, 3)$ are:

$AB$ and $BC$ are diagonals of adjacent faces of a rectangular box with its center at the origin,and its edges parallel to the coordinate axes. If the angles $\angle BOC, \angle COA$,and $\angle AOB$ are $\alpha, \beta$,and $\gamma$ respectively,then $\cos \alpha + \cos \beta + \cos \gamma$ is equal to:

The edge of a cube is of length $a$. Then the shortest distance between the diagonal of a cube and an edge skew to it is:

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:

The equation of the locus of a point whose distance from the $XY$-plane is twice its distance from the $Z$-axis is: