`1/(r_1^2)+1/(r_2^2)+1/(r_3^2)+1/(r^2)=`

Options

A. `(a^2+b^2+c^2)/(s^2)`
B. `(Sigma a^2)/(S^2)`
C. `4R`
D. `4R`

Answer

To solve the expression \( \frac{1}{r_1^2} + \frac{1}{r_2^2} + \frac{1}{r_3^2} + \frac{1}{r^2} \), where \( r_1, r_2, r_3 \) are the radii of the circumcircle and incircle of a triangle, we will use the following steps:

### Step 1: Understand the Variables
Let:
- \( r_1 \) be the circumradius (radius of the circumcircle).
- \( r_2 \) be the radius of the incircle.
- \( r_3 \) be the radius of the A-excircle (the excircle opposite to vertex A).
- \( r \) be the radius of the incircle.

### Step 2: Use the Formulas
We know the formulas for the circumradius \( R \) and the inradius \( r \):
- \( R = \frac{abc}{4S} \) where \( S \) is the area of the triangle, and \( a, b, c \) are the lengths of the sides of the triangle.
- \( r = \frac{S}{s} \) where \( s \) is the semi-perimeter, \( s = \frac{a + b + c}{2} \).

### Step 3: Substitute the Values
Now, we substitute the values of \( r_1, r_2, r_3 \) into the expression:
- \( r_1 = R \)
- \( r_2 = r \)
- \( r_3 \) can be expressed in terms of the triangle's sides and area.

### Step 4: Write the Expression
The expression becomes:
\[ \frac{1}{R^2} + \frac{1}{r^2} + \frac{1}{r_3^2} \]

### Step 5: Simplify the Expression
Using the known relationships:
- Substitute \( R \) and \( r \) into the expression.
- Combine the fractions using a common denominator.

### Step 6: Final Simplification
After combining and simplifying, we will arrive at a final expression that can be computed based on the triangle's sides and area.

### Final Result
The final result will be a simplified expression in terms of \( a, b, c \) and \( S \).