Let the incircle of `DeltaABC` touches the sides BC, CA, AB at `A_(1), B_(1),C_(1)` respectively. The incircle of `DeltaA_(1)B_(1)C_(1)` touches its sides of `B_(1)C_(1), C_(1)A_(1) and A_(1)B_(1)" at " A_(2), B_(2), C_(2)` respectively and so on.
Q. In `DeltaA_(4)B_(4)C_(4)`, the value of `angleA_(4)` is:

Options

A. `(3pi+A)/(6)`
B. `(3pi-A)/(8)`
C. `(5pi-A)/(16)`
D. `(5pi+A)/(16)`

Answer

To solve the problem step by step, we will analyze the relationship between the angles in the triangles formed by the incircles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a triangle \( \Delta ABC \) with an incircle that touches the sides at points \( A_1, B_1, C_1 \). The incircle of triangle \( \Delta A_1B_1C_1 \) touches its sides at points \( A_2, B_2, C_2 \), and this process continues to form triangles \( \Delta A_3B_3C_3 \) and \( \Delta A_4B_4C_4 \). We need to find the value of \( \angle A_4 \).

**Hint**: Draw the triangles and their incircles to visualize the problem better.

2. **Identifying Angles in Triangle \( \Delta ABC \)**:
The angle \( \angle A \) in triangle \( \Delta ABC \) is given. The angles at the points where the incircle touches the sides can be expressed in terms of \( \angle A \).

**Hint**: Remember that the angles formed at the center of the incircle are related to the angles of the triangle.

3. **Finding \( \angle A_1 \)**:
From the properties of the incircle, we know:
\[ \angle C_1OB_1 = \pi - \angle A \]
where \( O \) is the center of the incircle.

**Hint**: Use the fact that the angles around point \( O \) sum to \( \pi \).

4. **Finding \( \angle A_2 \)**:
In triangle \( \Delta OC_1A_1 \), we can write:
\[ \angle C_1OA_1 + 2 \times \angle OA_1C_1 = \pi \]
Since \( OC_1 = OA_1 \) (both are inradii), we can conclude:
\[ \angle A_1 = \frac{\pi - \angle C_1OA_1}{2} \]

**Hint**: Use the equality of angles in isosceles triangles.

5. **Finding \( \angle A_3 \)**:
Continuing this process, we find:
\[ \angle A_2 = \frac{\pi - \angle A_1}{2} = \frac{\pi - \frac{\pi - \angle A}{2}}{2} = \frac{3\pi + \angle A}{4} \]

**Hint**: Substitute the previously found angle into the new equation.

6. **Finding \( \angle A_4 \)**:
We can now find \( \angle A_4 \):
\[ \angle A_3 = \frac{\pi - \angle A_2}{2} = \frac{\pi - \frac{3\pi + \angle A}{4}}{2} = \frac{5\pi - \angle A}{8} \]
Finally, we find \( \angle A_4 \):
\[ \angle A_4 = \frac{\pi - \angle A_3}{2} = \frac{\pi - \frac{5\pi - \angle A}{8}}{2} = \frac{5\pi + \angle A}{16} \]

**Hint**: Keep track of the fractions carefully as they represent the angles.

### Final Answer:
The value of \( \angle A_4 \) is:
\[ \angle A_4 = \frac{5\pi + \angle A}{16} \]