Given the problem:

$\frac{a+b}{b+c} : \frac{b+c}{c+a} : \frac{c+a}{a+b} = 6 : 7 : 8$

and the information that $a + b + c = 14$.

To solve for $c$, let's analyze and work through the information provided.

1. Assign Ratios: Based on the given ratios, we can set up proportional equations:

   $\frac{a+b}{k} = 6, \quad \frac{b+c}{k} = 7, \quad \frac{c+a}{k} = 8$

   where $k$ is the common proportional constant.

2. Express $a+b$, $b+c$, and $c+a$ in terms of $k$: $a + b = 6k, \quad b + c = 7k, \quad c + a = 8k$

3. Use the Sum of Variables: We know that $(a + b) + (b + c) + (c + a) = 2(a + b + c) = 2 \cdot 14 = 28$

   Substituting our expressions in terms of $k$: $6k + 7k + 8k = 28$

4. Solve for $k$: $21k = 28 \Rightarrow k = \frac{28}{21} = \frac{4}{3}$

5. Calculate $a+b$, $b+c$, and $c+a$: $a + b = 6k = 6 \cdot \frac{4}{3} = 8$ $b + c = 7k = 7 \cdot \frac{4}{3} = \frac{28}{3} = 9.\overline{3}$ $c + a = 8k = 8 \cdot \frac{4}{3} = \frac{32}{3} = 10.\overline{6}$

6. Use $a + b + c = 14$ to solve for $c$**: From $a + b = 8$, we have: