RO-TST 2005 Problem 7

Let $A_0A_1A_2A_3A_4A_5$ be a convex hexagon inscribed in a circle. Define the points $A_0'$ , $A_2'$ , $A_4'$ on the circle, such that $$A_0A_0' \parallel A_2A_4, A_2A_2' \parallel A_4A_0, A_4A_4' \parallel A_2A_0 .$$

Let the lines $A_0'A_3$ and $A_2A_4$ intersect in $A_3'$ , the lines $A_2'A_5$ and $A_0A_4$ intersect in $A_5'$ and the lines $A_4'A_1$ and $A_0A_2$ intersect in $A_1'$ .

Prove that if the lines $A_0A_3$ , $A_1A_4$ and $A_2A_5$ are concurrent then the lines $A_0A_3'$ , $A_4A_1'$ and $A_2A_5'$ are also concurrent.