If the sides of a triangle are in A.P. as well as in G.P. then the value of `r_1/r_2 - r_2/r_3` is

Options

A. 1
B. 0
C. 2r
D. none

Answer

To solve the problem, we need to find the value of \( \frac{r_1}{r_2} - \frac{r_2}{r_3} \) given that the sides of a triangle are in both arithmetic progression (A.P.) and geometric progression (G.P.).

### Step-by-Step Solution:

1. **Define the sides of the triangle**:
Let the sides of the triangle be \( a, b, c \). Since they are in A.P., we can express them as:
\[ a = b - d, \quad b = b, \quad c = b + d \]
for some common difference \( d \).

2. **Condition for G.P.**:
Since the sides are also in G.P., we have:
\[ b^2 = ac \]
Substituting the values of \( a \) and \( c \):
\[ b^2 = (b - d)(b + d) = b^2 - d^2 \]
This implies:
\[ d^2 = 0 \implies d = 0 \]
Therefore, \( a = b = c \). This means the triangle is equilateral.

3. **Calculate the semi-perimeter \( S \)**:
The semi-perimeter \( S \) is given by:
\[ S = \frac{a + b + c}{2} = \frac{3b}{2} \]

4. **Calculate the area \( \Delta \)**:
For an equilateral triangle, the area \( \Delta \) can be calculated using the formula:
\[ \Delta = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} b^2 \]

5. **Calculate the circumradius \( R \)**:
The circumradius \( R \) of a triangle is given by:
\[ R = \frac{abc}{4\Delta} \]
For our equilateral triangle:
\[ R = \frac{b \cdot b \cdot b}{4 \cdot \frac{\sqrt{3}}{4} b^2} = \frac{b^3}{\sqrt{3} b^2} = \frac{b}{\sqrt{3}} \]

6. **Calculate the inradius \( r \)**:
The inradius \( r \) is given by:
\[ r = \frac{\Delta}{S} = \frac{\frac{\sqrt{3}}{4} b^2}{\frac{3b}{2}} = \frac{\sqrt{3} b}{6} \]

7. **Calculate \( \frac{r_1}{r_2} - \frac{r_2}{r_3} \)**:
Since \( r_1 \), \( r_2 \), and \( r_3 \) are the inradius, circumradius, and another radius respectively, we can assume \( r_1 = r \), \( r_2 = R \), and \( r_3 = R \) for an equilateral triangle.
Thus:
\[ \frac{r_1}{r_2} = \frac{\frac{\sqrt{3} b}{6}}{\frac{b}{\sqrt{3}}} = \frac{\sqrt{3}^2}{6} = \frac{3}{6} = \frac{1}{2} \]
and
\[ \frac{r_2}{r_3} = 1 \]
Therefore:
\[ \frac{r_1}{r_2} - \frac{r_2}{r_3} = \frac{1}{2} - 1 = -\frac{1}{2} \]

### Final Answer:
\[ \frac{r_1}{r_2} - \frac{r_2}{r_3} = -\frac{1}{2} \]