`a+b : b+c : c+a =6 : 7 : 8` find `a : b : c=?`

Answer

To solve the problem \( a + b : b + c : c + a = 6 : 7 : 8 \) and find the ratio \( a : b : c \), we can follow these steps:

### Step 1: Set up the equations
Given the ratios:
\[ a + b = 6k, \quad b + c = 7k, \quad c + a = 8k \]
where \( k \) is a common multiplier.

### Step 2: Add the equations
Now, we can add all three equations:
\[ (a + b) + (b + c) + (c + a) = 6k + 7k + 8k \]
This simplifies to:
\[ 2a + 2b + 2c = 21k \]
Dividing the entire equation by 2 gives:
\[ a + b + c = 10.5k \]

### Step 3: Solve for \( c \)
We know from the first equation that:
\[ c = (a + b + c) - (a + b) = 10.5k - 6k = 4.5k \]

### Step 4: Solve for \( a \)
Using the second equation:
\[ a = (a + b + c) - (b + c) = 10.5k - 7k = 3.5k \]

### Step 5: Solve for \( b \)
Using the third equation:
\[ b = (a + b + c) - (c + a) = 10.5k - 8k = 2.5k \]

### Step 6: Write the ratio
Now we have:
\[ a = 3.5k, \quad b = 2.5k, \quad c = 4.5k \]
Thus, the ratio \( a : b : c \) can be expressed as:
\[ a : b : c = 3.5k : 2.5k : 4.5k \]
Removing \( k \) from the ratio, we can write:
\[ a : b : c = 3.5 : 2.5 : 4.5 \]

### Step 7: Simplify the ratio
To simplify, we can multiply each part of the ratio by 2 to eliminate the decimals:
\[ a : b : c = 7 : 5 : 9 \]

### Final Answer
Thus, the final ratio is:
\[ \boxed{7 : 5 : 9} \]