Example 6 As shown in Figure 1, in the isosceles right triangle $AOB$ with one leg being 1, a point $P$ is randomly taken within this triangle. Parallel lines to the three sides are drawn through $P$, forming three triangles (shaded areas in the figure) with $P$ as a vertex. Find the minimum value of the sum of the areas of these three triangles and the position of point $P$ when the minimum value is achieved.