KöMaL 2006 K96

The sides of an equilateral triangle ABC are 16 cm long. The side $AC$ is extended beyond $A$ by one quarter of the length $AC$ and the point $P$ is obtained. The point $P$ is connected by a line to the point dividing the side $AB$ in a 1:3 ratio (so that the shorter segment of $AB$ lies next to $A$.) This line divides the side $BC$ into two pieces. Find their lengths.