If `r_(1),r_(2)andr_(3)` are exradii of any triangle , then `r_(1)r_(2)+r_(2)r_(3)+r_(3)r_(1)` is equal to

Options

A. `(Delta)/(r)`
B. `(Delta^(2))/(r^(2))`
C. `(r)/(Delta)`
D. `(r^(2))/(Delta^(2))`

Answer

To solve the problem, we need to find the value of \( r_1 r_2 + r_2 r_3 + r_3 r_1 \) in terms of the area \( \Delta \) of the triangle and its semi-perimeter \( s \).

### Step-by-Step Solution:

1. **Define the exradii**:
The exradii \( r_1, r_2, r_3 \) of a triangle can be expressed as:
\[ r_1 = \frac{\Delta}{s - a}, \quad r_2 = \frac{\Delta}{s - b}, \quad r_3 = \frac{\Delta}{s - c} \]
where \( a, b, c \) are the lengths of the sides of the triangle and \( s \) is the semi-perimeter given by \( s = \frac{a + b + c}{2} \).

2. **Calculate \( r_1 r_2 + r_2 r_3 + r_3 r_1 \)**:
We need to calculate:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 \]
Substituting the values of \( r_1, r_2, r_3 \):
\[ r_1 r_2 = \left(\frac{\Delta}{s - a}\right)\left(\frac{\Delta}{s - b}\right) = \frac{\Delta^2}{(s - a)(s - b)} \]
\[ r_2 r_3 = \left(\frac{\Delta}{s - b}\right)\left(\frac{\Delta}{s - c}\right) = \frac{\Delta^2}{(s - b)(s - c)} \]
\[ r_3 r_1 = \left(\frac{\Delta}{s - c}\right)\left(\frac{\Delta}{s - a}\right) = \frac{\Delta^2}{(s - c)(s - a)} \]
Therefore,
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{\Delta^2}{(s - a)(s - b)} + \frac{\Delta^2}{(s - b)(s - c)} + \frac{\Delta^2}{(s - c)(s - a)} \]

3. **Combine the fractions**:
The common denominator for these fractions is \( (s - a)(s - b)(s - c) \). Thus, we can write:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \Delta^2 \left( \frac{(s - c) + (s - a) + (s - b)}{(s - a)(s - b)(s - c)} \right) \]
Simplifying the numerator:
\[ (s - c) + (s - a) + (s - b) = 3s - (a + b + c) = 3s - 2s = s \]
Therefore, we have:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \Delta^2 \cdot \frac{s}{(s - a)(s - b)(s - c)} \]

4. **Express in terms of \( r \)**:
We know that the inradius \( r \) is given by:
\[ r = \frac{\Delta}{s} \]
Thus, \( \Delta = r \cdot s \). Substituting this into our expression:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = (r \cdot s)^2 \cdot \frac{s}{(s - a)(s - b)(s - c)} = \frac{r^2 s^3}{(s - a)(s - b)(s - c)} \]

5. **Final Result**:
Therefore, we conclude that:
\[ r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{r^2 s^3}{(s - a)(s - b)(s - c)} \]