AT-OEMO-REG 2003 Problem 3

Given are two parallel lines $g$ and $h$ and a point $P$ , that lies outside of the corridor bounded by $g$ and $h$ . Construct three lines $g_1$ , $g_2$ and $g_3$ through the point $P$ . These lines intersect $g$ in $A_1,A_2, A_3$ and $h$ in $B_1, B_2, B_3$ respectively. Let $C_1$ be the intersection of the lines $A_1B_2$ and $A_2B_1$ , $C_2$ be the intersection of the lines $A_1B_3$ and $A_3B_1$ and let $C_3$ be the intersection of the lines $A_2B_3$ and $A_3B_2$ . Show that there exists exactly one line $n$ , that contains the points $C_1,C_2,C_3$ and that $n$ is parallel to $g$ and $h$ .