if `(a)/(b + c) = (b)/(c + a) = (c)/(a + b)` and `a + b + c ne 0`, show that each given ratio is equal to `(1)/(2)`.

Answer

To solve the problem, we start with the given ratios:

\[ \frac{a}{b+c} = \frac{b}{c+a} = \frac{c}{a+b} \]

Let us denote this common ratio as \( k \). Therefore, we can write:

\[ \frac{a}{b+c} = k \quad (1) \]
\[ \frac{b}{c+a} = k \quad (2) \]
\[ \frac{c}{a+b} = k \quad (3) \]

From equation (1), we can express \( a \) in terms of \( k \):

\[ a = k(b+c) \quad (4) \]

From equation (2), we can express \( b \) in terms of \( k \):

\[ b = k(c+a) \quad (5) \]

From equation (3), we can express \( c \) in terms of \( k \):

\[ c = k(a+b) \quad (6) \]

Now, we can substitute equations (4), (5), and (6) into each other. Let's substitute \( a \) from (4) into (5):

\[ b = k\left(c + k(b+c)\right) \]

Expanding this gives:

\[ b = kc + k^2(b+c) \]

Rearranging terms, we have:

\[ b - k^2b = kc + k^2c \]

Factoring out \( b \) on the left side:

\[ b(1 - k^2) = c(k + k^2) \quad (7) \]

Now, let's substitute \( b \) from (5) into (6):

\[ c = k\left(k(c+a) + a\right) \]

Expanding this gives:

\[ c = k^2(c+a) + ka \]

Rearranging terms, we have:

\[ c - k^2c = k^2a + ka \]

Factoring out \( c \):

\[ c(1 - k^2) = a(k^2 + k) \quad (8) \]

Now we have two equations (7) and (8). We can express \( b \) and \( c \) in terms of \( a \) and \( k \):

From (7):

\[ b = \frac{c(k + k^2)}{1 - k^2} \]

From (8):

\[ c = \frac{a(k^2 + k)}{1 - k^2} \]

Now, we can substitute \( c \) back into the expression for \( b \):

Substituting \( c \) into (7):

\[ b(1 - k^2) = \frac{a(k^2 + k)(k + k^2)}{1 - k^2} \]

This leads to:

\[ b(1 - k^2)^2 = a(k^2 + k)(k + k^2) \]

Now we can sum \( a + b + c \):

\[ a + b + c = k(b+c) + k(c+a) + k(a+b) \]

This simplifies to:

\[ a + b + c = k(2(a + b + c)) \]

Since \( a + b + c \neq 0 \), we can divide both sides by \( a + b + c \):

\[ 1 = 2k \implies k = \frac{1}{2} \]

Thus, we have shown that:

\[ \frac{a}{b+c} = \frac{b}{c+a} = \frac{c}{a+b} = \frac{1}{2} \]

### Final Answer:
Each given ratio is equal to \( \frac{1}{2} \).