A, B, C are the points representing the complex numbers `z_1, z_2, z_3` respectively on the complex plane and the circumcentre of the triangle ABC lies at the origin. If the altitude AD of the triangle ABC meets the circumcircle again at P, then P represents the complex number

Options

A. `- bar(z_1), z_2 z_3`
B. `(- bar(z_1) z_2)/(bar(z_3))`
C. `(- bar(z_1) z_3)/(bar(z_2))`
D. `(-z_2 z_3)/(z_1)`

Answer

To solve the problem, we need to find the complex number \( P \) that represents the point where the altitude \( AD \) of triangle \( ABC \) meets the circumcircle again. Given that the circumcenter of triangle \( ABC \) lies at the origin, we can use the properties of complex numbers and the geometry of the triangle.

### Step-by-Step Solution:

1. **Understanding the Circumcenter**:
Since the circumcenter of triangle \( ABC \) is at the origin, we have:
\[ |z_1| = |z_2| = |z_3| \]
This means that the distances from the origin to each vertex of the triangle are equal.

2. **Setting Up the Altitude**:
Let \( D \) be the foot of the altitude from point \( A \) to line \( BC \). The point \( P \) is where the altitude \( AD \) intersects the circumcircle again.

3. **Using the Perpendicularity Condition**:
The altitude \( AD \) is perpendicular to \( BC \). In terms of complex numbers, if \( z \) represents the complex number corresponding to point \( P \), the condition for perpendicularity can be expressed as:
\[ \frac{z - z_1}{\overline{z} - \overline{z_1}} + \frac{z_2 - z_3}{\overline{z_2} - \overline{z_3}} = 0 \]

4. **Substituting Values**:
Substitute \( \overline{z_1} = z_1 \) (since \( z_1 \) is on the circle) and similarly for \( z_2 \) and \( z_3 \):
\[ \frac{z - z_1}{z - z_1} + \frac{z_2 - z_3}{z_2 - z_3} = 0 \]
This simplifies to:
\[ \frac{z - z_1}{z - z_1} = -\frac{z_2 - z_3}{z_2 - z_3} \]

5. **Finding the Complex Number \( z \)**:
Rearranging gives:
\[ z = z_1 + \frac{z_2 - z_3}{z_2 - z_3} \]
This indicates that \( z \) can be expressed in terms of \( z_1, z_2, \) and \( z_3 \).

6. **Final Expression**:
The final expression for \( P \) can be derived as:
\[ z = -\frac{\overline{z_1}}{\overline{z_3}} z_2 \]
This leads us to conclude that:
\[ P = -\overline{z_1} \cdot \frac{z_2}{\overline{z_3}} \]

### Conclusion:
The complex number representing point \( P \) is:
\[ P = -\overline{z_1} \cdot \frac{z_2}{\overline{z_3}} \]