2003 AMC 12A 第 16 题

The three triangles share equal bases, so their areas depend only on $P$’s distances to the sides

By the threefold symmetry each side yields the largest triangle equally often

The triangles $\triangle ABP,$ $\triangle ACP,$ and $\triangle BCP$ have equal bases (the sides of the equilateral triangle), so their areas are proportional to the distances from $P$ to those sides.

By the threefold symmetry of the equilateral triangle, $\triangle ABP$ is the largest with the same probability as each of the other two, so that probability is $\dfrac13.$

Thus, the correct answer is C.