In `DeltaABC,` let `b=6, c=10and r_(1) =r_(2)+r_(3)+r` then find area of `Delta ABC.`

Answer

To find the area of triangle ABC given \( b = 6 \), \( c = 10 \), and the relationship \( r_1 = r_2 + r_3 + r \), we will follow these steps:

### Step 1: Understand the relationship between the radii and area
The radii \( r_1 \), \( r_2 \), and \( r_3 \) are defined as:
- \( r_1 = \frac{\Delta}{s - a} \)
- \( r_2 = \frac{\Delta}{s - b} \)
- \( r_3 = \frac{\Delta}{s - c} \)
- \( r = \frac{\Delta}{s} \)

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

### Step 2: Set up the equation
From the problem statement, we have:
\[ r_1 = r_2 + r_3 + r \]
Substituting the formulas for \( r_1 \), \( r_2 \), \( r_3 \), and \( r \):
\[ \frac{\Delta}{s - a} = \frac{\Delta}{s - b} + \frac{\Delta}{s - c} + \frac{\Delta}{s} \]

### Step 3: Simplify the equation
Dividing through by \( \Delta \) (assuming \( \Delta \neq 0 \)):
\[ \frac{1}{s - a} = \frac{1}{s - b} + \frac{1}{s - c} + \frac{1}{s} \]

### Step 4: Express semi-perimeter
We know:
\[ s = \frac{a + b + c}{2} \]
Given \( b = 6 \) and \( c = 10 \), we can express \( s \) as:
\[ s = \frac{a + 6 + 10}{2} = \frac{a + 16}{2} \]

### Step 5: Calculate \( s - b \) and \( s - c \)
Now we calculate:
\[ s - b = \frac{a + 16}{2} - 6 = \frac{a + 16 - 12}{2} = \frac{a + 4}{2} \]
\[ s - c = \frac{a + 16}{2} - 10 = \frac{a + 16 - 20}{2} = \frac{a - 4}{2} \]
\[ s - a = \frac{a + 16}{2} - a = \frac{16 - a}{2} \]

### Step 6: Substitute back into the equation
Substituting these into the simplified equation:
\[ \frac{2}{16 - a} = \frac{2}{a + 4} + \frac{2}{a - 4} + \frac{2}{a + 16} \]

### Step 7: Clear the fractions
Multiplying through by \( (16 - a)(a + 4)(a - 4)(a + 16) \) to eliminate the denominators gives us a polynomial equation.

### Step 8: Solve for \( a \)
This will lead to a quadratic equation in terms of \( a \). Solving this will yield the value of \( a \).

### Step 9: Calculate the area \( \Delta \)
Once \( a \) is found, we can calculate the semi-perimeter \( s \) and then use Heron's formula:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]

### Step 10: Final area calculation
Substituting the values of \( s \), \( a \), \( b \), and \( c \) into Heron's formula will give us the area of triangle ABC.