`r_1+r_2+r_3-r=`

Options

A. R
B. 2R
C. 3R
D. 4R

Answer

To solve the expression \( r_1 + r_2 + r_3 - r \), where \( r_1, r_2, r_3 \) are the radii of the excircles of a triangle and \( r \) is the radius of the incircle, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Radii**:
- The radius of the excircle opposite to side \( a \) is given by:
\[ r_1 = \frac{\Delta}{s - a} \]
- The radius of the excircle opposite to side \( b \) is given by:
\[ r_2 = \frac{\Delta}{s - b} \]
- The radius of the excircle opposite to side \( c \) is given by:
\[ r_3 = \frac{\Delta}{s - c} \]
- The radius of the incircle is given by:
\[ r = \frac{\Delta}{s} \]

2. **Substitute the Radii into the Expression**:
- Substitute \( r_1, r_2, r_3, \) and \( r \) into the expression:
\[ r_1 + r_2 + r_3 - r = \frac{\Delta}{s - a} + \frac{\Delta}{s - b} + \frac{\Delta}{s - c} - \frac{\Delta}{s} \]

3. **Combine the Terms**:
- Factor out \( \Delta \):
\[ = \Delta \left( \frac{1}{s - a} + \frac{1}{s - b} + \frac{1}{s - c} - \frac{1}{s} \right) \]

4. **Find a Common Denominator**:
- The common denominator for the fractions is \( (s - a)(s - b)(s - c)s \).
- Rewrite each term with the common denominator:
\[ = \Delta \left( \frac{s(s - b)(s - c) + s(s - a)(s - c) + s(s - a)(s - b) - (s - a)(s - b)(s - c)}{(s - a)(s - b)(s - c)s} \right) \]

5. **Simplify the Numerator**:
- Expand and simplify the numerator:
- The expansion will yield a polynomial in terms of \( s \), \( a \), \( b \), and \( c \).
- After simplification, you will find that the numerator simplifies to \( 2s^2 - (a + b + c)s + ab + ac + bc \).

6. **Substituting \( a + b + c = 2s \)**:
- Substitute \( a + b + c = 2s \) into the expression:
\[ = \Delta \left( \frac{2s^2 - 2s^2 + ab + ac + bc}{(s - a)(s - b)(s - c)s} \right) \]
- This simplifies to:
\[ = \Delta \left( \frac{ab + ac + bc}{(s - a)(s - b)(s - c)s} \right) \]

7. **Final Expression**:
- The final expression can be related back to the circumradius \( R \) of the triangle:
\[ = \frac{abc}{4R} \]
- Thus, the expression \( r_1 + r_2 + r_3 - r \) simplifies to \( 4R \).

### Final Answer:
\[ r_1 + r_2 + r_3 - r = 4R \]