VN-TST 1999 Problem 3

Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$ , denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions:

I. $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$ . II. Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$ .

(By $(AB)$ we denote vector $AB$ )