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To solve this problem, let's start by analyzing the given information and the relationships provided:

### Given:
1. \( XY = 2 \).
2. \( AX = 2 \cdot YB \).
3. \( \frac{AZ}{ZC} = \frac{2}{3} \).

We are asked to find the ratio:

\[
\frac{[YBC]}{[ZYC]}.
\]

### Step 1: Analyze the Triangle and Given Ratios
Let's understand the given ratios and lengths:
- \( AX = 2 \cdot YB \) suggests that point \( X \) divides line segment \( AB \) such that the ratio \( \frac{AX}{YB} = 2 \).
- \( \frac{AZ}{ZC} = \frac{2}{3} \) means that point \( Z \) divides line segment \( AC \) such that the ratio \( \frac{AZ}{ZC} = \frac{2}{3} \).

### Step 2: Assign Variables to Segment Lengths
Let’s set up variables to represent the lengths of segments:
- Let \( AX = 2x \) and \( YB = x \), hence \( AB = AX + YB = 3x \).
- Let \( AZ = 2y \) and \( ZC = 3y \), hence \( AC = AZ + ZC = 5y \).

### Step 3: Use Area Ratios
Using the properties of similar triangles and area ratios:
1. Since \( X \) divides \( AB \) in the ratio \( 2:1 \), triangle \( \triangle AXC \) is similar to triangle \( \triangle BYC \), and triangle \( \triangle AXC \) is double the area of triangle \( \triangle BYC \).
  
2. Since \( Z \) divides \( AC \) in the ratio \( 2:3 \), triangle \( \triangle AZB \) is proportional to triangle \( \triangle ZBC \), and the areas of these triangles are in the ratio \( \frac{2}{3} \).

### Step 4: Calculate the Ratio \(\frac{[YBC]}{[ZYC]}\)
Given the similarity and area ratios:

\[
\frac{[YBC]}{[ZYC]} = \frac{\frac{1}{3} \times \frac{3}{5} \times [ABC]}{\frac{2}{3} \times [ABC]} = \frac{2}{5}.
\]

Thus, the solution to the ratio problem is:
\[
\boxed{\frac{[YBC]}{[ZYC]} = \frac{2}{5}}.
\]

Would you like further explanation or any other questions? 
Here are some related questions that you might be interested in:
1. How to derive the specific ratios of triangles from given points?
2. How to find area ratios given line segment division?
3. How does similar triangles relate to proportionality and ratios?
4. How to use given area properties and ratios to find other dimensions?
5. How does one determine unknown angles and triangle lengths based on provided information?

**Tip:** When solving for ratios involving line segments, always consider proportion and apply similar triangle and proportionality rules to make calculations easier.

In the diagram below, we have XY = 2, AX = 2 · YB, and AZ/ZC = 2/3. Find the ratio [YBC]/[ZYC].

Learn how to find the ratio of areas between two triangles formed by specific points and lines within a larger triangle. This problem covers essential geometric concepts, including the Triangle Similarity Theorem and Area Ratio Theorem. Ideal for students in Grades 10-12.

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