Suppose you have three circle touching each other mutually. What is the area of the triangle from by joining the centre of the circle.

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We have three circles that touch each other mutually. Each circle has its own radius: r1, r2, and r3. The centers of these circles are labeled C1, C2, and C3. When we connect these centers, they form a triangle. Our goal is to find the area of this triangle. To find the area of the triangle, we first need to determine its side lengths. Since the circles touch each other mutually, the distance between any two centers equals the sum of their radii. So side a equals r2 plus r3, side b equals r1 plus r3, and side c equals r1 plus r2. Now we apply Heron's formula to find the area. First, we calculate the semi-perimeter s, which equals a plus b plus c divided by 2. Substituting our values, we get s equals r1 plus r2 plus r3. Next, we use Heron's formula: Area equals the square root of s times s minus a times s minus b times s minus c. We calculate s minus a equals r1, s minus b equals r2, and s minus c equals r3. Now we substitute our values into Heron's formula. The area equals the square root of s times s minus a times s minus b times s minus c. This becomes the square root of r1 plus r2 plus r3, times r1, times r2, times r3. Rearranging, we get our final elegant formula: the area equals the square root of r1 times r2 times r3 times the sum r1 plus r2 plus r3. This beautiful result gives us the area of the triangle formed by the centers of three mutually touching circles. Let's verify our formula with a concrete example. Suppose we have three circles with radii 3, 4, and 5 units. Using our formula, the area equals the square root of 3 times 4 times 5 times the sum 3 plus 4 plus 5. This gives us the square root of 60 times 12, which equals the square root of 720, or 12 times the square root of 5, approximately 26.83 square units. This demonstrates the practical application of our elegant formula.