Three circles of radius a, b, c touch each other externally. The area of the triangle formed by joining their centre is

Options

A. `sqrt((a+b + c)abc)`
B. `(a+b+c)sqrt(ab+bc + ca)`
C. `ab + bc + ca`
D. None of the above

Answer

To find the area of the triangle formed by joining the centers of three circles with radii \(a\), \(b\), and \(c\) that touch each other externally, we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle formed by the centers of the circles are given by the distances between the centers:
- The distance between the centers of the circles with radii \(a\) and \(b\) is \(a + b\).
- The distance between the centers of the circles with radii \(b\) and \(c\) is \(b + c\).
- The distance between the centers of the circles with radii \(c\) and \(a\) is \(c + a\).

Thus, the sides of the triangle are:
- \(A = a + b\)
- \(B = b + c\)
- \(C = c + a\)

### Step 2: Calculate the semi-perimeter
The semi-perimeter \(s\) of the triangle is calculated as:
\[ s = \frac{A + B + C}{2} = \frac{(a + b) + (b + c) + (c + a)}{2} = \frac{2(a + b + c)}{2} = a + b + c \]

### Step 3: Apply Heron's formula
Heron's formula for the area \(A\) of a triangle with sides \(a\), \(b\), and \(c\) is given by:
\[ \text{Area} = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values of \(s\), \(A\), \(B\), and \(C\):
\[ \text{Area} = \sqrt{(a + b + c)((a + b + c) - (a + b))((a + b + c) - (b + c))((a + b + c) - (c + a))} \]

### Step 4: Simplify the expression
Calculating each term:
- \(s - A = (a + b + c) - (a + b) = c\)
- \(s - B = (a + b + c) - (b + c) = a\)
- \(s - C = (a + b + c) - (c + a) = b\)

Thus, we have:
\[ \text{Area} = \sqrt{(a + b + c)(c)(a)(b)} \]

### Final Result
The area of the triangle formed by the centers of the circles is:
\[ \text{Area} = \sqrt{(a + b + c)abc} \]