CZ-SK-MO 1957 Problem 2

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$ . Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$ . (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$ , where $X$ lies on line $VA$ and $Y$ lies on line $BC$ . Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$ . (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$