Given:
- Triangle ABC with sides AB, BC, and CA
- The sides AB, BC, and CA are extended to points D, E, and F respectively such that AB = BD, BC = CE, and CA = AF

To Find:
The ratio of the area of triangle ABC to the area of triangle DEF

Solution:

1. Identify Key Information:
- The sides AB, BC, and CA are extended to points D, E, and F respectively, creating triangle DEF.
- The sides AD, BE, and CF are parallel to the sides BC, CA, and AB respectively.
- The triangle ABC and triangle DEF share the same base, which is the line segment EF.

2. Identify Similar Triangles:
- Since AD || BC and BD || AC, triangle ADB and triangle ABC are similar.
- Similarly, triangle BEC and triangle ABC are similar, and triangle CFA and triangle ABC are similar.

3. Identify Ratios:
- From the similarity of triangle ADB and triangle ABC, we can identify that the ratio of the lengths of the corresponding sides is AD/AB = BD/BC = 1/2.
- Similarly, the ratio of the lengths of the corresponding sides in triangle BEC and triangle ABC is BE/BC = CE/CA = 1/2, and in triangle CFA and triangle ABC is CF/CA = AF/AB = 1/2.

4. Identify Ratios of Areas:
- The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides.
- Therefore, the ratio of the area of triangle ADB to the area of triangle ABC is (AD/AB)^2 = (1/2)^2 = 1/4.
- Similarly, the ratio of the area of triangle BEC to the area of triangle ABC is (BE/BC)^2 = (1/2)^2 = 1/4, and the ratio of the area of triangle CFA to the area of triangle ABC is (CF/CA)^2 = (1/2)^2 = 1/4.

5. Identify the Ratio of the Areas:
- The area of triangle ABC is the sum of the areas of triangle ADB, triangle BEC, and triangle CFA.
- Therefore, the ratio of the area of triangle ABC to the area of triangle DEF is:
(Area of triangle ABC) / (Area of triangle DEF) = [(Area of triangle ADB) + (Area of triangle BEC) + (Area of triangle CFA)] / (Area of triangle DEF)
= [(1/4) + (1/4) + (1/4)] / 1
= 3/4

Answer:
The ratio of the area of triangle ABC to the area of triangle DEF is 3:4.