VMO 1972 Problem 3

$ABC$ is a triangle. $U$ is a point on the line $BC$ . $I$ is the midpoint of $BC$ . The line through $C$ parallel to $AI$ meets the line $AU$ at $E$ . The line through $E$ parallel to $BC$ meets the line $AB$ at $F$ . The line through $E$ parallel to $AB$ meets the line $BC$ at $H$ . The line through $H$ parallel to $AU$ meets the line $AB$ at $K$ . The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$ . Show that $V, T$ and $I$ are collinear.