In trapezoid $abcd,$ base $\overline{ab}$ has length 6, and base $\overline{cd}$ has length 18. a line passes through the intersection of the diagonals, parallel to the bases. this line intersects $\overline{ad}$ and $\overline{bc}$ at $x$ and $y,$ respectively. find $xy.$ [asy] unitsize (1 cm); pair a, b, c, d, p, x, y; a = (4.5,3); b = (7,3); c = (8,0); d = (2,0); p = extension(a, c,b, d); x = extension(p, p + (1,0), a, d); y = extension(p, p + (1,0), b, c); draw(a--b--c--d--cycle); draw(a--c); draw(b--d); draw(x--y); label("$a$", a, nw); label("$b$", b, ne); label("$c$", c, se); label("$d$", d, sw); label("$x$", x, nw); label("$y$", y, ne); label("$6$", (a + b)/2, n); label("$18$", (c + d)/2, s); [/asy]