If in a triangle, `(1-(r_(1))/(r_(2))) (1 - (r_(1))/(r_(3))) = 2`, then the triangle is

Options

A. right angled
B. isosceles
C. equilateral
D. none of these

Explanation

To solve the problem, we need to analyze the given equation and use the properties of the triangle's inradii. The equation provided is:

\[ \left(1 - \frac{r_1}{r_2}\right) \left(1 - \frac{r_1}{r_3}\right) = 2 \]

Where:
- \( r_1 = \frac{\Delta}{s - a} \)
- \( r_2 = \frac{\Delta}{s - b} \)
- \( r_3 = \frac{\Delta}{s - c} \)

Here, \( \Delta \) is the area of the triangle, and \( s \) is the semi-perimeter given by \( s = \frac{a + b + c}{2} \).

### Step-by-Step Solution:

1. **Substitute \( r_1, r_2, r_3 \) into the equation**:
\[ 1 - \frac{r_1}{r_2} = 1 - \frac{\Delta/(s-a)}{\Delta/(s-b)} = 1 - \frac{s-b}{s-a} \]
\[ 1 - \frac{r_1}{r_3} = 1 - \frac{\Delta/(s-a)}{\Delta/(s-c)} = 1 - \frac{s-c}{s-a} \]

2. **Rewrite the equation**:
\[ \left(1 - \frac{s-b}{s-a}\right) \left(1 - \frac{s-c}{s-a}\right) = 2 \]

3. **Simplify each term**:
\[ 1 - \frac{s-b}{s-a} = \frac{s-a - (s-b)}{s-a} = \frac{b-a}{s-a} \]
\[ 1 - \frac{s-c}{s-a} = \frac{s-a - (s-c)}{s-a} = \frac{c-a}{s-a} \]

4. **Substituting back into the equation**:
\[ \left(\frac{b-a}{s-a}\right) \left(\frac{c-a}{s-a}\right) = 2 \]
\[ \frac{(b-a)(c-a)}{(s-a)^2} = 2 \]

5. **Cross-multiplying**:
\[ (b-a)(c-a) = 2(s-a)^2 \]

6. **Substituting \( s = \frac{a+b+c}{2} \)**:
\[ (b-a)(c-a) = 2\left(\frac{a+b+c}{2} - a\right)^2 \]
\[ = 2\left(\frac{b+c-a}{2}\right)^2 = \frac{(b+c-a)^2}{2} \]

7. **Expanding both sides**:
\[ (b-a)(c-a) = \frac{(b+c-a)^2}{2} \]

8. **Rearranging**:
\[ 2(b-a)(c-a) = (b+c-a)^2 \]

9. **Recognizing the condition**:
This condition is satisfied by a right-angled triangle, specifically when \( a^2 = b^2 + c^2 \).

### Conclusion:
Thus, the triangle is a **right-angled triangle**.