Find the point in the $XY$-plane which is equidistant from the three points $A(2, 0, 3)$,$B(0, 3, 2)$,and $C(0, 0, 1)$.

The centroid of a tetrahedron with vertices $P(5, -7, 0)$,$Q(a, 5, 3)$,$R(4, -6, b)$,and $S(6, c, 2)$ is $(4, -3, 2)$. Then the value of $2a + 3b + c$ is equal to:

Let $A(1, 8, 4)$, $B(0, -11, 3)$, and $C(2, -3, -1)$ be three points. If $D$ is the foot of the perpendicular from $A$ to the line $BC$, find the coordinates of $D$.

Consider the tetrahedron with the vertices $A(3,2,4)$,$B(x_1, y_1, 0)$,$C(x_2, y_2, 0)$,and $D(x_3, y_3, 0)$. If the triangle $BCD$ is formed by the lines $y=x$,$x+y=6$,and $y=1$,then the centroid of the tetrahedron is

The three different face diagonals of a cuboid (rectangular parallelepiped) have lengths $39, 40, 41$. The length of the main diagonal of the cuboid which joins a pair of opposite corners is

If $\alpha$ is the angle between any two diagonals of a cube and $\beta$ is the angle between a diagonal of a cube and a diagonal of its face, which intersects this diagonal of the cube, then $\cos \alpha + \cos^2 \beta =$