The user wants me to solve a multiple-choice question with multiple correct options regarding complex numbers.

Step 1: Analyze the given relation

We are given that $z_1$ and $z_2$ are distinct complex numbers and $z = (1-t)z_1 + tz_2$ for a real number $t$ with $0 < t < 1$. This equation is the parametric form of a line segment. Let's rearrange it to better understand the geometry.

$z - z_1 = (1-t)z_1 + tz_2 - z_1$ $z - z_1 = tz_2 - tz_1$ $z - z_1 = t(z_2 - z_1)$

This equation tells us that the complex number representing the vector from $z_1$ to $z$ (i.e., $z-z_1$) is a scalar multiple of the complex number representing the vector from $z_1$ to $z_2$ (i.e., $z_2-z_1$). Since $t$ is a real number, the points corresponding to $z_1, z_2,$ and $z$ in the Argand plane are collinear. Furthermore, since $0 < t < 1$, the point $z$ lies on the line segment joining $z_1$ and $z_2$, excluding the endpoints. The point $z$ divides the segment $z_1z_2$ in the ratio $t : (1-t)$.

Let's also find the expression for $z-z_2$: $z - z_2 = (1-t)z_1 + tz_2 - z_2$ $z - z_2 = (1-t)z_1 - (1-t)z_2$ $z - z_2 = (1-t)(z_1 - z_2) = -(1-t)(z_2 - z_1)$

Now, we will evaluate each option.

Step 2: Evaluate Option A

Option A is: $|z-z_1| + |z-z_2| = |z_1-z_2|$. This equation represents the triangle inequality becoming an equality, which is the condition for a point $z$ to lie on the line segment between $z_1$ and $z_2$. Since we've already established this geometrically, Option A must be correct.

Let's verify it algebraically: From Step 1, we have $z-z_1 = t(z_2-z_1)$. Taking the modulus: $|z-z_1| = |t(z_2-z_1)| = |t| |z_2-z_1|$. Since $0 < t < 1$, $|t|=t$. So, $|z-z_1| = t|z_2-z_1|$.

Also from Step 1, $z-z_2 = (1-t)(z_1-z_2)$. Taking the modulus: $|z-z_2| = |(1-t)(z_1-z_2)| = |1-t| |z_1-z_2|$. Since $0 < t < 1$, $1-t > 0$, so $|1-t|=1-t$. Thus, $|z-z_2| = (1-t)|z_1-z_2|$.

Adding the two results: $|z-z_1| + |z-z_2| = t|z_2-z_1| + (1-t)|z_1-z_2|$ Since $|z_2-z_1| = |z_1-z_2|$, we have: $|z-z_1| + |z-z_2| = (t + 1 - t)|z_1-z_2| = 1 imes |z_1-z_2| = |z_1-z_2|$. Thus, Option A is correct.

Step 3: Evaluate Option B

Option B is: $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z-z_2)$. From Step 1, we have: $z-z_1 = t(z_2-z_1)$ $z-z_2 = -(1-t)(z_2-z_1)$

Since $t$ is a positive real number, the vector $z-z_1$ has the same direction and argument as the vector $z_2-z_1$. Since $(1-t)$ is also a positive real number, $-(1-t)$ is a negative real number. This means the vector $z-z_2$ has the opposite direction to the vector $z_2-z_1$. Therefore, their arguments differ by $\pi$ (or $180^\circ$).

$\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z_2-z_1)$ $\operatorname{Arg}(z-z_2) = \operatorname{Arg}(-(1-t)(z_2-z_1)) = \operatorname{Arg}(-1) + \operatorname{Arg}(1-t) + \operatorname{Arg}(z_2-z_1) = \pi + 0 + \operatorname{Arg}(z_2-z_1)$. So, $\operatorname{Arg}(z-z_2) = \operatorname{Arg}(z-z_1) + \pi$ (modulo $2\pi$). The arguments are not equal. Thus, Option B is incorrect.

Step 4: Evaluate Option C

Option C is: $\left|\begin{array}{cc}z-z_1 & \bar{z}-\bar{z}_1 \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right|=0$. This determinant being zero is the standard condition for the collinearity of three points, here represented by $z, z_1, z_2$. The condition is equivalent to the ratio $\frac{z-z_1}{z_2-z_1}$ being a real number. Let's check the ratio using our result from Step 1: $\frac{z-z_1}{z_2-z_1} = \frac{t(z_2-z_1)}{z_2-z_1} = t$. Since $t$ is a real number, the condition is satisfied and the points are collinear. Therefore, the determinant must be zero.

Alternatively, we can substitute $z-z_1 = t(z_2-z_1)$ and $\bar{z}-\bar{z}_1 = \overline{t(z_2-z_1)} = t(\bar{z}_2-\bar{z}_1)$ (since $t$ is real) into the determinant: $\left|\begin{array}{cc}t(z_2-z_1) & t(\bar{z}_2-\bar{z}_1) \\ z_2-z_1 & \bar{z}_2-\bar{z}_1\end{array}\right| = t(z_2-z_1)(\bar{z}_2-\bar{z}_1) - t(\bar{z}_2-\bar{z}_1)(z_2-z_1) = 0$. Thus, Option C is correct.

Step 5: Evaluate Option D

Option D is: $\operatorname{Arg}(z-z_1) = \operatorname{Arg}(z_2-z_1)$. From Step 1, we have $z-z_1 = t(z_2-z_1)$. Since $t$ is a positive real number ($0 < t < 1$), the complex number $z-z_1$ is obtained by multiplying the complex number $z_2-z_1$ by a positive real scalar. This operation scales the magnitude but does not change the argument. Geometrically, the vector from $z_1$ to $z$ points in the same direction as the vector from $z_1$ to $z_2$. Therefore, their arguments must be equal. Thus, Option D is correct.

Conclusion

The correct options are A, C, and D.