Let $z_1$ and $z_2$ be two distinct complex numbers and let $z = (1 - t) z_1 + t z_2$ for some real number $t$ with $0 < t < 1$. If $\text{Arg}(w)$ denotes the principal argument of a non-zero complex number $w$, then

Let z₁ and z₂ be two distinct complex numbers and let z = (1 - t) z₁ + t z₂ for some real number t with 0 < t < 1. If Arg(w) denotes the principal argument of a non-zero complex number w, then

$z = (1 - t)z_1 + t z_2, \quad 0 < t < 1$

$\text{Arg}(w) = \text{principal argument of } w \neq 0$

$\text{Which statements (1)--(4) are TRUE?}$

$z = (1 - t)z_1 + t z_2, \quad 0 < t < 1$

$\Rightarrow z \text{ is an interior point of the segment } z_1 z_2$

$a = z_2 - z_1, \quad z - z_1 = t\,a, \quad z - z_2 = -(1 - t)\,a$

$|z - z_1| + |z - z_2| = t|a| + (1 - t)|a|$

$= |a| = |z_1 - z_2|$

$\therefore \text{(1) is TRUE}$

$\text{Arg}(z - z_1) = \text{Arg}(a)$

$\text{Arg}(z - z_2) = \text{Arg}\big({-(1 - t)a}\big) = \text{Arg}(a) \pm \pi$

$\therefore \text{(2) is FALSE}$

z - z₁ z̄ - z̄₁ z₂ - z₁ z̄₂ - z̄₁

$= (z - z_1)(\bar{z}_2 - \bar{z}_1) - (\bar{z} - \bar{z}_1)(z_2 - z_1)$

$= (t a)\bar{a} - (t \bar{a})a = t|a|^2 - t|a|^2$

$= 0 \quad\Rightarrow\quad \text{(3) is TRUE}$

$\text{Arg}(z - z_1) = \text{Arg}(t a) = \text{Arg}(a) = \text{Arg}(z_2 - z_1)$

$\therefore \text{(4) is TRUE}$

$\boxed{\text{Correct statements: (1), (3), (4)}}$