Step 1: Understanding the Concept:
We are given ratios of sums of sides of a triangle. By setting these ratios equal to a constant $k$, we can solve a simple system of linear equations to find the relative proportions of the individual sides $a, b,$ and $c$. Once the side proportions are known, we can utilize the Cosine Rule to find the ratios of the cosines of the angles.
Step 2: Key Formula or Approach:
1. Express sides $a, b, c$ in terms of a constant $k$. Sum the equations to find $a+b+c$, then subtract individual pairs to isolate $a, b$, and $c$.
2. Cosine Rule: $\cos A = \frac{b^2+c^2-a^2}{2bc}$, $\cos B = \frac{a^2+c^2-b^2}{2ac}$, $\cos C = \frac{a^2+b^2-c^2}{2ab}$.
Step 3: Detailed Explanation:
Let the given ratio equal $k$:
\[ \frac{b+c}{11} = \frac{c+a}{12} = \frac{a+b}{13} = k \]
This gives a system of three equations:
1) $b + c = 11k$
2) $c + a = 12k$
3) $a + b = 13k$
Add all three equations together:
\[ 2a + 2b + 2c = 11k + 12k + 13k = 36k \]
Divide by 2:
\[ a + b + c = 18k \]
Now, subtract the initial equations from this sum to find individual sides:
$a = (a+b+c) - (b+c) = 18k - 11k = 7k$
$b = (a+b+c) - (c+a) = 18k - 12k = 6k$
$c = (a+b+c) - (a+b) = 18k - 13k = 5k$
Since we only need ratios of cosines, we can drop the constant $k$ and just use the relative lengths: $a=7, b=6, c=5$.
Now apply the Cosine Rule for each angle:
\[ \cos A = \frac{b^2+c^2-a^2}{2bc} = \frac{6^2+5^2-7^2}{2(6)(5)} = \frac{36+25-49}{60} = \frac{12}{60} = \frac{1}{5} \]
\[ \cos B = \frac{a^2+c^2-b^2}{2ac} = \frac{7^2+5^2-6^2}{2(7)(5)} = \frac{49+25-36}{70} = \frac{38}{70} = \frac{19}{35} \]
\[ \cos C = \frac{a^2+b^2-c^2}{2ab} = \frac{7^2+6^2-5^2}{2(7)(6)} = \frac{49+36-25}{84} = \frac{60}{84} = \frac{5}{7} \]
We need the ratio $\cos A : \cos B : \cos C$.
Ratio $= \frac{1}{5} : \frac{19}{35} : \frac{5}{7}$
To remove fractions, multiply by the least common multiple of the denominators, which is 35:
\[ \left(\frac{1}{5} \times 35\right) : \left(\frac{19}{35} \times 35\right) : \left(\frac{5}{7} \times 35\right) \]
\[ 7 : 19 : 25 \]
Step 4: Final Answer:
The ratio is $7 : 19 : 25$.