Three circles touch one-anothet externally and tangent at their point of contact meet at a point. If their radii be `r_1 , r_2 , r_3` then the distance of this point from eithet ot their points of contact is ……

Answer

To find the distance from the point where three circles touch each other externally to either of their points of contact, we can use the following steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- Let the three circles have radii \( r_1, r_2, r_3 \).
- The centers of the circles will be denoted as \( O_1, O_2, O_3 \).
- The point where the circles touch each other externally is denoted as \( P \).

2. **Finding the Distances Between Centers**:
- The distance between the centers of the circles can be calculated as:
\[ O_1O_2 = r_1 + r_2, \quad O_2O_3 = r_2 + r_3, \quad O_3O_1 = r_3 + r_1 \]

3. **Calculating the Semi-Perimeter**:
- The semi-perimeter \( s \) of triangle \( O_1O_2O_3 \) is given by:
\[ s = \frac{O_1O_2 + O_2O_3 + O_3O_1}{2} = \frac{(r_1 + r_2) + (r_2 + r_3) + (r_3 + r_1)}{2} \]
- Simplifying this, we get:
\[ s = \frac{2(r_1 + r_2 + r_3)}{2} = r_1 + r_2 + r_3 \]

4. **Calculating the Area of Triangle**:
- The area \( \Delta \) of triangle \( O_1O_2O_3 \) can be calculated using Heron's formula:
\[ \Delta = \sqrt{s(s - O_1O_2)(s - O_2O_3)(s - O_3O_1)} \]
- Substituting the lengths:
\[ \Delta = \sqrt{s \left(s - (r_1 + r_2)\right) \left(s - (r_2 + r_3)\right) \left(s - (r_3 + r_1)\right)} \]
- This simplifies to:
\[ \Delta = \sqrt{(r_1 + r_2 + r_3) \left(r_3\right) \left(r_1\right) \left(r_2\right)} \]

5. **Finding the Inradius**:
- The inradius \( r \) of triangle \( O_1O_2O_3 \) is given by:
\[ r = \frac{\Delta}{s} \]
- Substituting the values we found:
\[ r = \frac{\sqrt{(r_1 + r_2 + r_3) \cdot r_1 \cdot r_2 \cdot r_3}}{r_1 + r_2 + r_3} \]

6. **Final Expression**:
- The distance from the point \( P \) to either of the points of contact is equal to the inradius \( r \):
\[ r = \sqrt{\frac{r_1 \cdot r_2 \cdot r_3}{r_1 + r_2 + r_3}} \]