Olympiad Number Theory · Superchallenge problem

Let $A = (0, 0)$, $B = (6, 0)$, $C = (2, 4)$. The point $P$ with barycentric coordinates $(\tfrac12, \tfrac13, \tfrac16)$ with respect to $(A, B, C)$ has Cartesian coordinates $(x, y)$. Compute $x + y$.