Tetrahedron $ABCD$ has side lengths $AB = CD = 12$. These edges are perpendicular to each other. Let $E$ and $F$ be the midpoints of $AB$ and $CD$ respectively. Given that $EF = 10$ and $EF$ is perpendicular to both $AB$ and $CD$,find the volume of the tetrahedron $ABCD$.

$A$ square $ABCD$ of diagonal $2a$ is folded along the diagonal $AC$ so that the planes $DAC$ and $BAC$ are at a right angle. The shortest distance between $DC$ and $AB$ is

The length of the internal bisector of angle $A$ in $\triangle ABC$ with vertices $A(4,7,8)$,$B(2,3,4)$ and $C(2,5,7)$ 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

The incenter of the triangle $ABC$,whose vertices are $A(0,2,1)$,$B(-2,0,0)$,and $C(-2,0,2)$ is

$\Pi_1, \Pi_2, \Pi_3$ 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