Every circle has form |z-a|=k|z-b|

[Grothard]

We can express any circle in the complex plane as |z-a|=k|z-b| where a and b are distinct complex numbers, k > 0 and $k \not= 1.$

Is there an elegant way of showing this fundamental property of the complex plane to be true?

 

[mathman]

It doesn't look elegant, but squaring both sides leads to a straightforward proof.

 

[AlephZero]

Google for "circle of Apollonius". This geometrical result was known centuries before complex numbers and analytic geometry were invented.