`a+b+c : b+c+d : c+d+a : d+a+b = 6:7:8:9`
Find `a:b:c:d=?`

Answer

To solve the problem, we need to find the values of \(a\), \(b\), \(c\), and \(d\) based on the given ratios \(a+b+c : b+c+d : c+d+a : d+a+b = 6:7:8:9\).

### Step-by-Step Solution:

1. **Set Up the Equations:**
Let:
- \(a + b + c = 6k\)
- \(b + c + d = 7k\)
- \(c + d + a = 8k\)
- \(d + a + b = 9k\)

Here, \(k\) is a common multiplier.

2. **Add All Equations:**
Adding all four equations:
\[ (a+b+c) + (b+c+d) + (c+d+a) + (d+a+b) = 6k + 7k + 8k + 9k \]
This simplifies to:
\[ 3a + 3b + 3c + 3d = 30k \]
Dividing the entire equation by 3 gives:
\[ a + b + c + d = 10k \]

3. **Find Individual Variables:**
Now we can express \(d\) in terms of \(k\):
- From \(a + b + c = 6k\), we can substitute \(d\):
\[ d = (a + b + c + d) - (a + b + c) = 10k - 6k = 4k \]

- From \(b + c + d = 7k\):
\[ b + c + 4k = 7k \implies b + c = 3k \]

- From \(c + d + a = 8k\):
\[ c + 4k + a = 8k \implies a + c = 4k \]

- From \(d + a + b = 9k\):
\[ 4k + a + b = 9k \implies a + b = 5k \]

4. **Solve for \(a\), \(b\), and \(c\):**
Now we have three equations:
- \(a + b = 5k\) (1)
- \(b + c = 3k\) (2)
- \(a + c = 4k\) (3)

From equation (1), we can express \(b\):
\[ b = 5k - a \]

Substitute \(b\) into equation (2):
\[ (5k - a) + c = 3k \implies c = 3k - 5k + a = -2k + a \]

Substitute \(c\) back into equation (3):
\[ a + (-2k + a) = 4k \implies 2a - 2k = 4k \implies 2a = 6k \implies a = 3k \]

5. **Find \(b\) and \(c\):**
Substitute \(a = 3k\) back into the equations:
- From (1):
\[ 3k + b = 5k \implies b = 2k \]
- From (2):
\[ 2k + c = 3k \implies c = k \]

6. **Find \(d\):**
We already found \(d\) earlier:
\[ d = 4k \]

7. **Final Ratios:**
Now we have:
- \(a = 3k\)
- \(b = 2k\)
- \(c = k\)
- \(d = 4k\)

Therefore, the ratio \(a : b : c : d\) is:
\[ 3k : 2k : k : 4k = 3 : 2 : 1 : 4 \]

### Conclusion:
The final answer is:
\[ a : b : c : d = 3 : 2 : 1 : 4 \]