With usual notation in `Delta ABC,` the numerical value of
`((a+b+c)/(r_(1)+r_(2)+r_(3))) ((a)/(r_(1))+(b)/(r _(2))+ (c)/(r_(3)))` is

Answer

To solve the problem, we need to find the numerical value of the expression:

\[ \frac{(a+b+c)}{(r_1 + r_2 + r_3)} \left( \frac{a}{r_1} + \frac{b}{r_2} + \frac{c}{r_3} \right) \]

where \( r_1, r_2, r_3 \) are the exradii of triangle \( ABC \).

### Step 1: Define the exradii
The exradii \( r_1, r_2, r_3 \) can be expressed in terms of the area \( \Delta \) of triangle \( ABC \) and the semi-perimeter \( S \):
- \( r_1 = \frac{\Delta}{S - a} \)
- \( r_2 = \frac{\Delta}{S - b} \)
- \( r_3 = \frac{\Delta}{S - c} \)

### Step 2: Substitute the exradii into the expression
Now we substitute \( r_1, r_2, r_3 \) into the expression:

\[ r_1 + r_2 + r_3 = \frac{\Delta}{S - a} + \frac{\Delta}{S - b} + \frac{\Delta}{S - c} \]

### Step 3: Simplify the denominator
To simplify \( r_1 + r_2 + r_3 \):

\[ r_1 + r_2 + r_3 = \Delta \left( \frac{1}{S - a} + \frac{1}{S - b} + \frac{1}{S - c} \right) \]

Finding a common denominator, we get:

\[ r_1 + r_2 + r_3 = \Delta \cdot \frac{(S - b)(S - c) + (S - a)(S - c) + (S - a)(S - b)}{(S - a)(S - b)(S - c)} \]

### Step 4: Substitute into the main expression
Now substituting back into the main expression:

\[ \frac{(a+b+c)}{r_1 + r_2 + r_3} \left( \frac{a}{r_1} + \frac{b}{r_2} + \frac{c}{r_3} \right) \]

### Step 5: Simplify the first part
The first part becomes:

\[ \frac{(a+b+c)}{\Delta \cdot \frac{(S - b)(S - c) + (S - a)(S - c) + (S - a)(S - b)}{(S - a)(S - b)(S - c)}} \]

### Step 6: Simplify the second part
The second part can be simplified as follows:

\[ \frac{a}{r_1} + \frac{b}{r_2} + \frac{c}{r_3} = \frac{a(S - a)}{\Delta} + \frac{b(S - b)}{\Delta} + \frac{c(S - c)}{\Delta} \]

This simplifies to:

\[ \frac{1}{\Delta} \left( a(S - a) + b(S - b) + c(S - c) \right) \]

### Step 7: Combine both parts
Combining both parts, we get:

\[ \frac{(a+b+c)}{\Delta} \cdot \frac{(a(S - a) + b(S - b) + c(S - c))}{(S - b)(S - c) + (S - a)(S - c) + (S - a)(S - b)} \]

### Step 8: Evaluate the expression
After simplifying and substituting \( S = \frac{a+b+c}{2} \), we can evaluate the expression. Ultimately, through simplification, we find that the numerical value of the entire expression is:

\[ \boxed{4} \]