Certainly! Here is a test case generator for the Extended Euclid Algorithm problem:

```python
import random

def tcgen():
    # Generate two positive integers a and b within the given constraints
    a = random.randint(1, 10**9)
    b = random.randint(1, 10**9)
    
    return a, b

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

### Explanation:

1. **Generating `a` and `b`**:
   - We use `random.randint(1, 10**9)` to generate random integers `a` and `b` in the range `[1, 10^9]`.

2. **Returning the Test Case**:
   - The function returns a tuple of the two generated integers.

3. **Example Usage**:
   - An example usage is provided to demonstrate how to generate and print a test case.

This generator will provide random pairs of integers `a` and `b` within the given constraints, ready to be used for testing implementations of the Extended Euclid Algorithm problem.