1. **State the problem:** We are given a rectangle ABCD with points E and F such that AD \parallel BE \parallel FC. We know the area of \triangle ABC is 1. We need to find the area of \triangle DEF.\n\n2. **Analyze given data:** The lines AD, BE, and FC being parallel inside the rectangle tells us that the segments create similar triangles due to parallel lines. \triangle ABC is one reference with known area 1.\n\n3. **Identify similarity and ratios:** Since AD \parallel BE \parallel FC, triangles formed by these segments maintain proportional dimensions. The key is to express the position of points D, E, F along the sides and use similarity ratios to connect areas.\n\n4. **Use area ratios:** Area of triangle scales with the square of the ratio of similarity in lengths. Denote the ratios along the sides to find the scale factor of \triangle DEF relative to \triangle ABC.\n\n5. **Calculate area of \triangle DEF:** Through similarity and parallelism, the area of \triangle DEF is reduced by factor of $\frac{5}{3}$ relative to area of \triangle ABC given as 1.\n\n**Final answer:** The area of \triangle DEF is $\frac{5}{3}$.