KöMaL 2013 C1195

Let $a$, $b$ and $c$, respectively, be the lengths of the face diagonals $AF$, $FC$and $CA$ of a cuboid ABCDEFGH . Let $\alpha$, $\beta$ and $\gamma$, respectively, denote the angles enclosed by the face diagonals with the diagonal $BH$ of the cuboid. Prove that $a^.$cos $\alpha$-$b^.$cos $\beta$+$c^.$cos $\gamma$=0 if the length of edge $AB$ is between the lengths of the edges $BC$ and $BF$.