Consider a sequence $P_1, P_2, \ldots$ of points in the plane such that $P_1, P_2, P_3$ are non-collinear and for every $n \geq 4$, $P_n$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_1$ and $P_5$. Prove the following:
(a) The area of the triangle formed by the points $P_n, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity.
(b) The point $P_9$ lies on $L$.