Internal bisectors of `DeltaABC` meet the circumcircle at point D, E, and F
Area of `DeltaDEF` is

Options

A. `2R^(2) cos^(2) ((A)/(2)) cos^(2)((B)/(2)) cos^(2) ((C)/(2))`
B. `2R^(2) sin ((A)/(2)) sin ((B)/(2)) sin((C)/(2))`
C. `2R^(2) sin^(2) ((A)/(2)) sin^(2) ((B)/(2) sin^(2) ((C)/(2))`
D. `2R^(2) cos((A)/(2)) cos((B)/(2)) cos((C)/(2))`

Explanation

To find the area of triangle DEF, formed by the internal bisectors of triangle ABC meeting the circumcircle, we can follow these steps:

### Step 1: Understand the Geometry
We have triangle ABC with its internal angle bisectors intersecting the circumcircle at points D, E, and F. We need to find the area of triangle DEF.

### Step 2: Use the Area Formula for Triangle ABC
The area of triangle ABC can be expressed using the formula:
\[ \text{Area}_{ABC} = \frac{abc}{4R} \]
where \( a, b, c \) are the lengths of the sides opposite to angles A, B, and C respectively, and R is the circumradius of triangle ABC.

### Step 3: Determine the Angles at Points D, E, and F
Using the properties of angle bisectors:
- Angle ABE = \( \frac{B}{2} \)
- Angle ACD = \( \frac{C}{2} \)
- Angle FDE = \( \frac{B + C}{2} \)

### Step 4: Apply the Sine Rule in Triangle DEF
By applying the sine rule in triangle DEF, we can express the sides in terms of the angles:
\[ \frac{EF}{\sin(FDE)} = 2R \]
Thus, we can find:
\[ EF = 2R \cdot \sin(FDE) \]

### Step 5: Calculate the Area of Triangle DEF
The area of triangle DEF can be calculated using the formula:
\[ \text{Area}_{DEF} = \frac{1}{2} \times EF \times h \]
where \( h \) is the height from point D to line EF.

Using the sine of the angles:
\[ \text{Area}_{DEF} = 2R^2 \cdot \sin\left(\frac{A+B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \cdot \sin\left(\frac{A+C}{2}\right) \]

### Step 6: Simplify the Area Expression
Using the product-to-sum identities, we can simplify the area expression:
\[ \text{Area}_{DEF} = 2R^2 \cdot \cos\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right) \cdot \cos\left(\frac{C}{2}\right) \]

### Final Result
Thus, the area of triangle DEF is:
\[ \text{Area}_{DEF} = 2R^2 \cdot \cos\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right) \cdot \cos\left(\frac{C}{2}\right) \]

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