Sure! Below is a test case generator for the given "Extended Euclid Algorithm" problem statement:

### Test Case Generator for Extended Euclid Algorithm

```python
import random

def extended_euclid_tcgen():
    # Generate two large random positive integers a and b
    a = random.randint(1, 10**9)
    b = random.randint(1, 10**9)
    
    return a, b

# Example of generating a test case
a, b = extended_euclid_tcgen()
print(f"{a} {b}")
```

### Explanation:

- **Random Integers Generation**: We generate two random integers \(a\) and \(b\) within the range \([1, 10^9]\), as specified by the problem constraints.
  
- **Return Values**: The function returns the generated integers \(a\) and \(b\) formatted as expected by the problem statement.

### Example Output:

Running the provided generator might output something like:

```
837462345 192837465
```

This will provide a set of diverse test cases that can be used to validate the correctness of the implemented Extended Euclid Algorithm.