Question

If $z_1,z_2,z_3,z_4$ are the affixes of four points in the Argand plane. z is affix of a point, then prove that $z_1,z_2,z_3,z_4$ are concyclic.

Answer 1

We have,
$|z-z_1|=|z-z_2|=|z-z_3|=|z-z_4|$
Therefore, the point having affix $z$ is equidistant from the four point having affixes $z_1,z_2,z_3,z_4$. Thus,, $z$ is the affix of either the center of a circle or the point of intersection of diagonals of a square. Therefore, $z_1,z_2,z_3,z_4$ are concyclic.
Ans: 1