The image presents a geometric problem involving a triangle with a line parallel to one of its sides and some area relations. Here’s a breakdown of the key details from the diagram:

• DE || AC: Line DE is parallel to line AC.
• BD:AB = 1:4: The ratio of segments BD to AB is 1:4.
• S_ADEC = 60: The area of triangle ADEC is 60.
• S_ABC = x: The area of triangle ABC is unknown, denoted by x.

Since DE is parallel to AC, triangles ABE and ABC are similar by the basic proportionality theorem (also known as Thales’ theorem). This means that the ratio of the areas of these triangles will be the square of the ratio of corresponding sides.

Now, let's compute the unknown area, S_ABC.

Given the ratio of segments BD:AB is 1:4, we can conclude that:

• $\frac{BD}{AB} = \frac{1}{4}$

This also means the ratio of areas of the smaller triangle ABE to the larger triangle ABC will be the square of this ratio:

• $\frac{S_{ABE}}{S_{ABC}} = \left(\frac{1}{4}\right)^2 = \frac{1}{16}$

Since the total area of triangle ABC, S_ABC, consists of the areas of triangles ABE and ADEC, we have:

• $S_{ABC} = S_{ABE} + S_{ADEC}$

Substitute the known value of S_ADEC = 60 and express S_ABE as $\frac{1}{16}S_{ABC}$:

$S_{ABC} = \frac{1}{16}S_{ABC} + 60$

Now, solve for S_ABC (x): $S_{ABC} - \frac{1}{16}S_{ABC} = 60$ $\frac{15}{16}S_{ABC} = 60$ $S_{ABC} = 60 \times \frac{16}{15} = 64$

Thus, the area of triangle ABC, $S_{ABC}$, is 64.

Would you like to explore more details or need further clarification?

Here are some related questions:

1. How does the area ratio between similar triangles relate to the ratio of corresponding sides?
2. How can Thales' theorem be applied in other geometric situations?
3. What is the significance of parallel lines in triangle geometry?
4. How can you calculate areas when given side ratios and similarity?
5. What other theorems are useful for solving geometric problems involving similar triangles?

Tip: When working with similar triangles, remember that the ratio of areas is the square of the ratio of corresponding sides!