If two triangles whose vertices are respectively the complex numbers `z_(1),z_(2),z_(3)` and `a_(1),a_(2),a_(3)` are similar, then the determinant.
`|{:(z_(1),a_(1),1),(z_(2),a_(2),1),(z_(3),a_(3),1):}|` is equal to

Options

A. `z_(1)z_(2)z_(3)`
B. `a_(1)a_(2)a_(3)`
C. 1
D. 0

Explanation

To solve the problem, we need to show that if two triangles with vertices represented by complex numbers \( z_1, z_2, z_3 \) and \( a_1, a_2, a_3 \) are similar, then the determinant

\[ \left| \begin{array}{ccc} z_1 & a_1 & 1 \\ z_2 & a_2 & 1 \\ z_3 & a_3 & 1 \end{array} \right| = 0. \]

### Step-by-Step Solution:

1. **Understanding Similar Triangles**:
- Two triangles are similar if their corresponding angles are equal and the ratios of their corresponding sides are equal.

2. **Setting Up the Ratios**:
- For the triangles to be similar, we can express the ratios of the sides in terms of the complex numbers:
\[ \frac{z_1 - z_2}{z_3 - z_2} = \frac{a_1 - a_2}{a_3 - a_2}. \]
- This means that the direction and magnitude of the sides are proportional.

3. **Cross Multiplying**:
- Cross-multiplying the above equation gives us:
\[ (z_1 - z_2)(a_3 - a_2) = (a_1 - a_2)(z_3 - z_2). \]

4. **Rearranging**:
- Rearranging this equation leads to:
\[ z_1 (a_3 - a_2) - z_2 (a_3 - a_2) = a_1 (z_3 - z_2) - a_2 (z_3 - z_2). \]

5. **Forming the Determinant**:
- We can express this relationship in terms of a determinant:
\[ \left| \begin{array}{ccc} z_1 & a_1 & 1 \\ z_2 & a_2 & 1 \\ z_3 & a_3 & 1 \end{array} \right| = 0. \]

6. **Conclusion**:
- Since the determinant equals zero, it indicates that the points represented by the complex numbers \( z_1, z_2, z_3 \) and \( a_1, a_2, a_3 \) are collinear, which is consistent with the property of similar triangles.

### Final Result:
Thus, we conclude that the determinant is equal to zero:
\[ \left| \begin{array}{ccc} z_1 & a_1 & 1 \\ z_2 & a_2 & 1 \\ z_3 & a_3 & 1 \end{array} \right| = 0. \]