I have the following exercise:
Let $z_1$ and $z_2$ be complex numbers. Prove that the complex number $z_3 =
\frac{z_1 + z_2}{2}$ is the midpoint between $z_1$ and $z_2$.
My attempt:
It is easy to prove that $|z_1 – z_3| – |z_2 – z_3| = 0$, which implies that the distance between $z_1$ and $z_2$ is the same that the distance between $z_2$ and $z_3$. Is it sufficient to prove that $|z_1 – z_3| + |z_2 – z_3| = |z_1 – z_2|$?
Thank you in advance.
Best Answer
One deifinition of mid-point is the following: $z_3$ is the mid point of $z_1$ and $z_2$ if it lies on the line joining $z_1$ and $z_2$ and is equidistant from these points. You have proved the second property. To prove the first property note that $(z_1-z_3)=(z_3-z_2)$. This implies that $z_3$ lies on the line joining $z_1$ and $z_2$.