`(1/r - 1/r_1)(1/r-1/r_2)(1/r - 1/r_3)=`

Options

A. `(16R)/(r^2(Sigma a)^2)`
B. `R/((Sigma a)^2)`
C. `(4R)/((Sigma a)^2)`
D. `(16R)/((Sigma a)^2)`

Answer

To solve the expression \((\frac{1}{r} - \frac{1}{r_1})(\frac{1}{r} - \frac{1}{r_2})(\frac{1}{r} - \frac{1}{r_3})\), we will follow these steps:

### Step 1: Define the variables
We know that:
- \( r_1 \) is the radius of the incircle of the triangle.
- \( r_2 \) is the radius of the A-excircle.
- \( r_3 \) is the radius of the B-excircle.
- \( r \) is the radius of the circumcircle.

### Step 2: Use the formulas for the radii
The formulas for the radii are:
- \( r_1 = \frac{\Delta}{s} \)
- \( r_2 = \frac{\Delta}{s - a} \)
- \( r_3 = \frac{\Delta}{s - b} \)
- \( r = \frac{abc}{4\Delta} \)

Where:
- \( \Delta \) is the area of the triangle.
- \( s \) is the semi-perimeter, given by \( s = \frac{a + b + c}{2} \).

### Step 3: Substitute the values into the expression
We can rewrite the expression as:
\[ \left(\frac{1}{r} - \frac{s}{\Delta}\right)\left(\frac{1}{r} - \frac{s - a}{\Delta}\right)\left(\frac{1}{r} - \frac{s - b}{\Delta}\right) \]

### Step 4: Simplify each term
Substituting the values of \( r_1, r_2, r_3 \):
1. For \( \frac{1}{r} - \frac{1}{r_1} \):
\[ \frac{1}{r} - \frac{s}{\Delta} = \frac{\Delta - sr}{r\Delta} \]
2. For \( \frac{1}{r} - \frac{1}{r_2} \):
\[ \frac{1}{r} - \frac{s - a}{\Delta} = \frac{\Delta - (s - a)r}{r\Delta} \]
3. For \( \frac{1}{r} - \frac{1}{r_3} \):
\[ \frac{1}{r} - \frac{s - b}{\Delta} = \frac{\Delta - (s - b)r}{r\Delta} \]

### Step 5: Combine the terms
Now we multiply the three simplified terms:
\[ \left(\frac{\Delta - sr}{r\Delta}\right)\left(\frac{\Delta - (s - a)r}{r\Delta}\right)\left(\frac{\Delta - (s - b)r}{r\Delta}\right) \]

### Step 6: Factor out common terms
Factoring out \( \frac{1}{(r\Delta)^3} \):
\[ \frac{1}{(r\Delta)^3} \left(\Delta - sr\right)\left(\Delta - (s - a)r\right)\left(\Delta - (s - b)r\right) \]

### Step 7: Final expression
The final expression can be simplified further depending on the values of \( a, b, c \) and their relationships.