Three circles of radius a, b, c touch each other externally. The area of triangle formed by joining their centers is.
a, b,c त्रिज्याओं वाले तीन वृत्त एक दूसरे को बाहर से स्पर्श करते हैं। उनके केन्द्रों को जोड़ने से बने त्रिभुज का क्षेत्रफल है| Options are (a) `sqrt((a+b+c)abc)` (b)`(a+b+c) sqrt(ab+bc+ca)` (c)`ab+bc+ca` (d)None of these.

Options

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

Explanation

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

### Step 1: Identify the lengths of the sides of the triangle
The distances between the centers of the circles are equal to the sum of their radii. Therefore, the lengths of the sides of the triangle formed by the centers of the circles are:
- Side \(AB = a + b\)
- Side \(BC = b + c\)
- Side \(CA = c + a\)

### Step 2: Use Heron's formula to find the area of the triangle
To find the area of the triangle, we can use Heron's formula, which states that the area \(A\) of a triangle with sides \(x\), \(y\), and \(z\) is given by:
\[ A = \sqrt{s(s-x)(s-y)(s-z)} \]
where \(s\) is the semi-perimeter of the triangle:
\[ s = \frac{x+y+z}{2} \]

In our case:
- \(x = a + b\)
- \(y = b + c\)
- \(z = c + a\)

Calculating the semi-perimeter \(s\):
\[ s = \frac{(a+b) + (b+c) + (c+a)}{2} = \frac{2(a+b+c)}{2} = a + b + c \]

### Step 3: Substitute into Heron's formula
Now we can substitute \(s\), \(x\), \(y\), and \(z\) into Heron's formula:
\[ A = \sqrt{(a+b+c)((a+b+c)-(a+b))((a+b+c)-(b+c))((a+b+c)-(c+a))} \]
This simplifies to:
\[ A = \sqrt{(a+b+c)(c)(a)(b)} \]

### Step 4: Final expression for the area
Thus, the area of the triangle formed by the centers of the circles is:
\[ A = \sqrt{(a+b+c)abc} \]

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