AUSTRIAN-POLISH 1995 Problem 5

$ABC$ is an equilateral triangle. $A_{1}, B_{1}, C_{1}$ are the midpoints of $BC, CA, AB$ respectively. $p$ is an arbitrary line through $A_{1}$ . $q$ and $r$ are lines parallel to $p$ through $B_{1}$ and $C_{1}$ respectively. $p$ meets the line $B_{1}C_{1}$ at $A_{2}$ . Similarly, $q$ meets $C_{1}A_{1}$ at $B_{2}$ , and $r$ meets $A_{1}B_{1}$ at $C_{2}$ . Show that the lines $AA_{2}, BB_{2}, CC_{2}$ meet at some point $X$ , and that $X$ lies on the circumcircle of $ABC$ .