Three circles with radius `r_(1), r_(2), r_(3)` touch one another externally. The tangents at their point of contact meet at a point whose distance from a point of contact is `2`. The value of `((r_(1)r_(2)r_(3))/(r_(1)+r_(2)+r_(3)))` is equal to

Explanation

To solve the problem, we will follow the steps outlined in the video transcript, ensuring we derive the required expression step by step.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
We have three circles with radii \( r_1, r_2, r_3 \) that touch each other externally. The points of contact form a triangle with vertices at the centers of the circles.

2. **Identifying the Distances**:
The distances between the centers of the circles are:
- Distance between Circle 1 and Circle 2: \( r_1 + r_2 \)
- Distance between Circle 1 and Circle 3: \( r_1 + r_3 \)
- Distance between Circle 2 and Circle 3: \( r_2 + r_3 \)

3. **Setting Up the Triangle**:
Let the points of contact be \( A, B, C \) corresponding to circles with radii \( r_1, r_2, r_3 \) respectively. The triangle formed by these points is denoted as triangle \( ABC \).

4. **Incenter and Radius**:
The tangents at the points of contact meet at a point \( O \) (the incenter of triangle \( ABC \)), and the distance from \( O \) to any side of the triangle is given as 2. This means the inradius \( r \) of triangle \( ABC \) is 2.

5. **Calculating the Semi-perimeter**:
The semi-perimeter \( s \) of triangle \( ABC \) is given by:
\[ s = \frac{(r_2 + r_3) + (r_1 + r_3) + (r_1 + r_2)}{2} = r_1 + r_2 + r_3 \]

6. **Area of Triangle**:
The area \( \Delta \) of triangle \( ABC \) can be expressed using the inradius:
\[ \Delta = r \cdot s = 2 \cdot (r_1 + r_2 + r_3) \]

7. **Using Heron's Formula**:
According to Heron's formula, the area can also be expressed as:
\[ \Delta = \sqrt{s(s - (r_2 + r_3))(s - (r_1 + r_3))(s - (r_1 + r_2))} \]
Substituting the values:
\[ \Delta = \sqrt{(r_1 + r_2 + r_3)(r_1)(r_2)(r_3)} \]

8. **Equating the Two Area Expressions**:
Setting the two expressions for area equal gives:
\[ 2(r_1 + r_2 + r_3) = \sqrt{(r_1 + r_2 + r_3)(r_1)(r_2)(r_3)} \]

9. **Squaring Both Sides**:
Squaring both sides results in:
\[ 4(r_1 + r_2 + r_3)^2 = (r_1 + r_2 + r_3)(r_1)(r_2)(r_3) \]

10. **Dividing by \( (r_1 + r_2 + r_3) \)**:
Assuming \( r_1 + r_2 + r_3 \neq 0 \), we can divide both sides by \( (r_1 + r_2 + r_3) \):
\[ 4(r_1 + r_2 + r_3) = (r_1)(r_2)(r_3) \]

11. **Rearranging the Equation**:
Rearranging gives:
\[ \frac{(r_1)(r_2)(r_3)}{(r_1 + r_2 + r_3)} = 4 \]

### Final Result:
Thus, the value of the expression \( \frac{r_1 r_2 r_3}{r_1 + r_2 + r_3} \) is equal to **4**.