`A(z_1)`,`B(z_2)` and `C(z_3)` are the vertices of triangle ABC inscribed in the circle |z|=2,internal angle bisector of angle A meets the circumcircle again at `D(z_4)`.Point D is:

Options

A. `z_(4) = (1)/(z_(2))+(1)/(z_(3))`
B. `sqrt((z_(2) +z_(3))/(z_(1))`
C. `sqrt((z_(2)z_(3))/(z_(1))`
D. `z_(4) = sqrt(z_(2)z_(3))`

Explanation

To solve the problem, we need to find the point D, which is the intersection of the internal angle bisector of angle A with the circumcircle of triangle ABC. The vertices of the triangle are represented by complex numbers \( z_1 \), \( z_2 \), and \( z_3 \), and the circumcircle has a radius of 2.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- The triangle ABC is inscribed in the circle of radius 2, which means that \( |z_1| = |z_2| = |z_3| = 2 \).
- The internal angle bisector of angle A meets the circumcircle again at point D.

2. **Properties of the Angle Bisector**:
- The internal angle bisector divides the opposite side in the ratio of the adjacent sides. In terms of complex numbers, if \( D \) lies on the circumcircle and is the intersection of the angle bisector, then the angles \( \angle BAD \) and \( \angle CAD \) are equal.

3. **Using the Argument Property**:
- Since \( D \) lies on the circumcircle, we can express the argument of \( z_4 \) (which represents point D) in terms of the arguments of \( z_2 \) and \( z_3 \):
\[ \text{arg}(z_4) = \frac{\text{arg}(z_2) + \text{arg}(z_3)}{2} \]

4. **Magnitude of Points**:
- Since \( z_1, z_2, z_3 \) lie on the circle of radius 2, we have:
\[ |z_1| = |z_2| = |z_3| = 2 \]
- Thus, we can express \( |z_2| \) and \( |z_3| \) as:
\[ |z_2| = 2 \quad \text{and} \quad |z_3| = 2 \]

5. **Finding \( z_4 \)**:
- The property of the angle bisector in terms of complex numbers states that:
\[ z_4 = \sqrt{z_2 z_3} \]
- This is derived from the fact that the angle bisector divides the angle into two equal parts, and thus the point D can be represented as the geometric mean of \( z_2 \) and \( z_3 \).

6. **Final Expression**:
- Therefore, we conclude that:
\[ z_4 = \sqrt{z_2 z_3} \]

### Conclusion:
The point D is given by the expression:
\[ z_4 = \sqrt{z_2 z_3} \]