# Geometric interpretation of $|\frac{z+i} {z-i}| =2$

Consider the equation

$$\left|\frac{z+i} {z-i}\right| =2$$

Solving it yields a circle, but I wonder if the equation itself has a geometric interpretation.

• It's the points that are twice as far from $-i$ as from $i$. Commented Dec 9, 2017 at 6:43
• It is a circle of Apollonius. See this answer for the general case.
– dxiv
Commented Dec 9, 2017 at 6:46
• @Vrisk: When posting a question, the body of the post should be self-contained; if relevant information appears in the title, it should be duplicated in the body of the post as well.
– user14972
Commented Apr 6, 2018 at 11:29
• Also, I've rewritten your question to make it more clear; check to make sure I haven't changed your meaning.
– user14972
Commented Apr 6, 2018 at 11:29

From $|\frac{z+i} {z-i}| = 2$, we get $\frac{|z – (–i)|} {|z – (i)|} = 2$.

If we let P, A, B to represent the complex numbers z, +i, and -i respectively, we have $\frac{BP} {AP} = \frac 21$.

This means we have another point C lying on AB such that PC is the angle bisector of $\angle APB$. For details, see the “angle bisector theorem”.

The same is true for the existence of another point D such that PD is the external angle bisector of $\angle APB$.

Note that (1) the angle between the internal and external angle bisector of the same angle is $\frac {\pi}{2}$; and (2) C, D are fixed points on AB. Hence, P lies on the circle with CD as diameter.

• Simply beautiful, thank you! Commented Dec 10, 2017 at 11:01

Here is a sketch of what geometrically is going on:

NOTE

As AB is constant, the equation describes a circle known as the Circle of Apollonius.

https://en.wikipedia.org/wiki/Circles_of_Apollonius