Let z₁ and z₂, be two distinct complex numbers and let $\mathrm{z}=(1-\mathrm{t}){\mathrm{z}}_1+t z_2$ for some real number t with 0 <  t < 1. if arg (w) denotes the principal argument of a non-zero complex number , then

Given $\mathrm{z}=\frac{(1-\mathrm{t}){\mathrm{z}}_1+{\mathrm{tz}}_2}{(1-\mathrm{t})+\mathrm{t}}$

Clearly, z divides z, and z, in the ratio of t :(1 - t), 0<t<1

$\Rightarrow \mathrm{AP}+\mathrm{BP}=\mathrm{AB}\text{ i.e., }\left|\mathrm{z}-{\mathrm{z}}_1\right|+\left|\mathrm{z}-{\mathrm{z}}_2\right|=\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|.$

and $\arg \left(\mathrm{z}-{\mathrm{z}}_1\right)=\arg \left({\mathrm{z}}_2-\mathrm{z}\right)=\arg \left({\mathrm{z}}_2-{\mathrm{z}}_1\right)$

Also ,$\arg \left(\mathrm{z}-{\mathrm{z}}_1\right)=\arg \left({\mathrm{z}}_2-{\mathrm{z}}_1\right)$

$\Rightarrow \arg \left(\frac{\mathrm{z}-{\mathrm{z}}_1}{{\mathrm{z}}_2-{\mathrm{z}}_1}\right)=0$

$\therefore \frac{z-z_1}{z_2-z_1}$ is purely real.

$\Rightarrow \frac{\mathrm{z}-{\mathrm{z}}_1}{{\mathrm{z}}_2-{\mathrm{z}}_1}=\frac{\overline{\mathrm{z}}-{\overline{\mathrm{z}}}_1}{{\overline{\mathrm{z}}}_2-{\overline{\mathrm{z}}}_1}\text{ or }\left|\begin{array}{cc}\mathrm{z}-{\mathrm{z}}_1& \overline{\mathrm{z}}-{\overline{\mathrm{z}}}_1\\ {\mathrm{z}}_2-{\mathrm{z}}_1& {\overline{\mathrm{z}}}_2-{\overline{\mathrm{z}}}_1\end{array}\right|=0$