Internal angle bisecotors of `DeltaABC` meets its circum circle at D, E and F where symbols have usual meaning.
Q. The ratio of area of triangle ABC and triangle DEF is :

Options

A. `ge1`
B. `le1`
C. `ge 1//2`
D. `le1//2`

Answer

To find the ratio of the area of triangle ABC to the area of triangle DEF, we can follow these steps:

### Step 1: Area of Triangle ABC
The area of triangle ABC can be expressed using the formula:
\[ \text{Area}_{ABC} = \frac{1}{2} \times AB \times AC \times \sin C \]
Alternatively, using the circumradius \( R \):
\[ \text{Area}_{ABC} = \frac{abc}{4R} \]
where \( a, b, c \) are the lengths of the sides opposite to angles A, B, and C respectively.

### Step 2: Area of Triangle DEF
The angles at points D, E, and F can be determined using the properties of angle bisectors:
- Angle D = \(\frac{B + C}{2}\)
- Angle E = \(\frac{A + C}{2}\)
- Angle F = \(\frac{A + B}{2}\)

Using the circumradius \( R \) for triangle DEF, the area can be expressed as:
\[ \text{Area}_{DEF} = \frac{DEF}{4R} = \frac{(DE)(EF)(FD)}{4R} \]

### Step 3: Relationship Between Angles
Using the sine function, we can express the area of triangle DEF in terms of the angles:
\[ \text{Area}_{DEF} = 2R^2 \sin D \sin E \sin F \]
Substituting the angles:
- \(\sin D = \sin\left(\frac{B + C}{2}\right)\)
- \(\sin E = \sin\left(\frac{A + C}{2}\right)\)
- \(\sin F = \sin\left(\frac{A + B}{2}\right)\)

### Step 4: Ratio of Areas
Now, we can find the ratio of the areas:
\[ \frac{\text{Area}_{ABC}}{\text{Area}_{DEF}} = \frac{2R^2 \sin A \sin B \sin C}{2R^2 \sin D \sin E \sin F} \]
This simplifies to:
\[ \frac{\sin A \sin B \sin C}{\sin D \sin E \sin F} \]

### Step 5: Simplifying the Ratio
Using the identities for sine, we can express:
\[ \sin D = \cos\left(\frac{A}{2}\right), \quad \sin E = \cos\left(\frac{B}{2}\right), \quad \sin F = \cos\left(\frac{C}{2}\right) \]
Thus, the ratio becomes:
\[ \frac{\sin A \sin B \sin C}{\cos\left(\frac{A}{2}\right) \cos\left(\frac{B}{2}\right) \cos\left(\frac{C}{2}\right)} \]

### Conclusion
The final ratio of the area of triangle ABC to triangle DEF is:
\[ \frac{\text{Area}_{ABC}}{\text{Area}_{DEF}} = 8 \cdot \frac{\sin\left(\frac{A}{2}\right) \sin\left(\frac{B}{2}\right) \sin\left(\frac{C}{2}\right)} \]