Similar Triangles: Analyzing DBC and ABC Triangle Properties

To solve this problem, we need to use the properties of similar triangles. It is given that triangles DBC and ABC are similar. When triangles are similar, it means that their corresponding sides are proportional.

For triangles DBC and ABC, considering the similarity, the corresponding sides should satisfy:

• $\frac{AB}{BD} = \frac{AC}{CD} = \frac{BC}{BC}$

The ratio $\frac{BC}{BC}$ simplifies to 1, which is a characteristic of the proportional relationship between the sides of similar triangles.

To solve for the correct choice, let's compare this with the options provided:

• Option 1: $\frac{AD}{AB} = \frac{AE}{AD} = \frac{BC}{BC}$ - This does not align because segments AD and AE aren't involved in the main similarity relation.
• Option 2: $\frac{AB}{BD} = \frac{AC}{CD} = \frac{BC}{BC}$ - This perfectly aligns with our derived relationship.
• Option 3: $\frac{AD}{DB} = \frac{AE}{EC} = \frac{BC}{BC}$ - This option has segments that are not directly part of the triangles ABC and DBC as described.
• Option 4: $\frac{AD}{AB} = \frac{AE}{AC} = \frac{BC}{BC}$ - Again, this involves segments not described in the initial triangle similarity.

Therefore, the correct correspondence that mathematically represents the similarity of the given triangles is found in Option 2.

Hence, the correct relation of similarity is: $\frac{AB}{BD} = \frac{AC}{CD} = \frac{BC}{BC}$.