Three circles whose radii are 2, 3, 4 units and having centres as `C_1,C_2,C_3` respectively touch each other externally at D,E,F. The circumradius of triangle DEF is :

Options

A. `(2sqrt6)/3`
B. `(4sqrt6)/3`
C. `(sqrt6)/3`
D. None

Answer

To find the circumradius of triangle DEF formed by the points of contact of three circles with radii 2, 3, and 4 units, we can follow these steps:

### Step 1: Determine the lengths of the sides of triangle ABC
The centers of the circles are C1, C2, and C3, with corresponding radii r1 = 2, r2 = 3, and r3 = 4. The sides of triangle ABC can be calculated as follows:
- Side AB = r1 + r2 = 2 + 3 = 5
- Side BC = r2 + r3 = 3 + 4 = 7
- Side AC = r1 + r3 = 2 + 4 = 6

Thus, the sides of triangle ABC are:
- AB = 5
- BC = 7
- AC = 6

### Step 2: Calculate the semi-perimeter (s) of triangle ABC
The semi-perimeter (s) is given by:
\[ s = \frac{AB + BC + AC}{2} = \frac{5 + 7 + 6}{2} = \frac{18}{2} = 9 \]

### Step 3: Calculate the area (Δ) of triangle ABC using Heron's formula
Heron's formula states:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
Where a, b, and c are the lengths of the sides of the triangle.
Here, we have:
- a = 7 (BC)
- b = 6 (AC)
- c = 5 (AB)

Now substituting the values:
\[ \Delta = \sqrt{9(9-7)(9-6)(9-5)} \]
\[ \Delta = \sqrt{9 \times 2 \times 3 \times 4} \]
\[ \Delta = \sqrt{9 \times 24} \]
\[ \Delta = \sqrt{216} \]
\[ \Delta = 6\sqrt{6} \]

### Step 4: Calculate the circumradius (R) of triangle ABC
The circumradius (R) is given by the formula:
\[ R = \frac{abc}{4\Delta} \]
Where a, b, and c are the lengths of the sides of the triangle.
Substituting the values:
\[ R = \frac{5 \times 6 \times 7}{4 \times 6\sqrt{6}} \]
\[ R = \frac{210}{24\sqrt{6}} \]
\[ R = \frac{35}{4\sqrt{6}} \]
To rationalize the denominator:
\[ R = \frac{35\sqrt{6}}{24} \]

### Conclusion
The circumradius of triangle DEF is:
\[ R = \frac{35\sqrt{6}}{24} \]