, 2014

There are given four rays, $a$, $b$, $c$ and $d$ in space, starting from the same point, laying in a plane $\varPi$. For an arbitrary acute angle $\varphi$, rotate $\varPi$ by angle $\varphi$ in positive direction around each of the four rays; denote the rotated planes by $A_\varphi$, $B_\varphi$, $\varGamma_\varphi$ and $\varDelta_\varphi$, respectively. Let $\varSigma_\varphi$ be the plane through the intersection line of $A_\varphi$ and $B_\varphi$, and the intersection line of $\varGamma_\varphi$ and $\varDelta_\varphi$. Show that the planes $\varSigma_\varphi$ share a common line.
(5 pont)