`z_(1),z_(2),z_(3)` and `z'_(1),z'_(2),z'_(3)` are nonzero complex numbers such that `z_(3) = (1-lambda)z_(1) + lambdaz_(2)` and `z'_(3) = (1-mu)z'_(1) + mu z'_(2)`, then which of the following statements is/are ture?

Options

A. If `lambda, mu in R -{0},` then `z_(1),z_(2)` and `z_(3)` are colliner and `z_(1),z_(2),z_(3)` are colliner separately.
B. If `lambda,mu` are complex numbers, where `lambda = mu`, then triangles formed by points `z_(1),z_(2),z_(3)` and `z'_(1),z'_(2),z'_(3)` are similare.
C. If `lambda, mu` are distinct complex numbers, then points `z_(1),z_(2),z_(3)` and `z'_(1),z'_(2),z_(3)` are not connectd by any well defined gemetry.
D. If `0 lt lambda lt 1`, then `z_(3)` divides the line joining `z_(1)` and `z_(2)` internally and if `mu gt1`, then `z_(3)` divides the following of `z'_(1),z'_(2)` extranlly

Explanation

To solve the problem, we need to analyze the given complex numbers and their relationships. Let's break down the solution step by step.

### Step 1: Understand the Given Conditions
We have two sets of complex numbers:
- \( z_3 = (1 - \lambda) z_1 + \lambda z_2 \)
- \( z'_3 = (1 - \mu) z'_1 + \mu z'_2 \)

Here, \( \lambda \) and \( \mu \) are parameters that determine the position of \( z_3 \) and \( z'_3 \) on the line segment joining \( z_1 \) and \( z_2 \) (for \( z_3 \)) and \( z'_1 \) and \( z'_2 \) (for \( z'_3 \)).

### Step 2: Analyze the Position of \( z_3 \)
The expression for \( z_3 \) indicates that it divides the line segment joining \( z_1 \) and \( z_2 \) in the ratio \( \lambda : (1 - \lambda) \).

**Hint:** If \( 0 < \lambda < 1 \), then \( z_3 \) lies between \( z_1 \) and \( z_2 \) (internally). If \( \lambda < 0 \) or \( \lambda > 1 \), then \( z_3 \) lies outside the segment (externally).

### Step 3: Analyze the Position of \( z'_3 \)
Similarly, for \( z'_3 \):
- The expression \( z'_3 = (1 - \mu) z'_1 + \mu z'_2 \) shows that \( z'_3 \) divides the line segment joining \( z'_1 \) and \( z'_2 \) in the ratio \( \mu : (1 - \mu) \).

**Hint:** The same conditions apply for \( \mu \) as for \( \lambda \).

### Step 4: Determine Collinearity
Since \( z_3 \) and \( z'_3 \) are defined in terms of \( z_1, z_2 \) and \( z'_1, z'_2 \), we can say that if \( \lambda \) and \( \mu \) are both real and between 0 and 1, then the points \( z_1, z_2, z_3 \) and \( z'_1, z'_2, z'_3 \) are collinear.

**Hint:** Collinearity can be established by showing that the ratios of the segments formed by these points are equal.

### Step 5: Compare Arguments
If \( \lambda = \mu \), then we can compare the arguments of the complex numbers:
- The argument of the ratio \( \frac{z_3 - z_1}{z_2 - z_1} \) will equal the argument of the ratio \( \frac{z'_3 - z'_1}{z'_2 - z'_1} \).

**Hint:** This leads to the conclusion that triangles formed by these points are similar.

### Conclusion
Based on the analysis, we can conclude that:
- If \( \lambda \) and \( \mu \) are both real and within the interval (0, 1), then \( z_3 \) and \( z'_3 \) divide their respective segments internally.
- If \( \lambda \) or \( \mu \) are outside this interval, the divisions are external.
- If \( \lambda = \mu \), the triangles formed by the points are similar.

Thus, all statements regarding the relationships of these complex numbers based on the conditions provided are true.

### Final Statements
- The points \( z_1, z_2, z_3 \) are collinear.
- The points \( z'_1, z'_2, z'_3 \) are collinear.
- The triangles formed by these points are similar if \( \lambda = \mu \).