If sides of triangle ABC are a, b, and c such that `2b = a + c`, then

Options

A. `(b)/(c) gt (2)/(3)`
B. `(b)/(c) gt (1)/(3)`
C. `(b)/(c) lt 2`
D. `(b)/(c) lt (3)/(2)`

Explanation

To solve the problem where the sides of triangle ABC are given as \( a, b, \) and \( c \) such that \( 2b = a + c \), we will derive inequalities involving the sides of the triangle.

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 2b = a + c \]

2. **Rearrange the equation to express \( a \) in terms of \( b \) and \( c \):**
\[ a = 2b - c \]

3. **Apply the triangle inequality:**
The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. We will apply this to find relationships between \( a, b, \) and \( c \).

4. **First inequality: \( a + b > c \)**
Substitute \( a = 2b - c \) into the inequality:
\[ (2b - c) + b > c \]
Simplifying this gives:
\[ 3b - c > c \implies 3b > 2c \implies \frac{b}{c} > \frac{2}{3} \]

5. **Second inequality: \( b + c > a \)**
Again, substitute \( a = 2b - c \):
\[ b + c > (2b - c) \]
Simplifying this gives:
\[ b + c > 2b - c \implies 2c > b \implies \frac{b}{c} < 2 \]

6. **Combine the results:**
From the inequalities derived, we have:
\[ \frac{2}{3} < \frac{b}{c} < 2 \]

### Conclusion:
The inequalities indicate that the ratio \( \frac{b}{c} \) is bounded between \( \frac{2}{3} \) and \( 2 \).