FP2 Complex loci help!

The point P represents the complex number z, where
|z+i| = sqrt 2 |z+i/2| .
Show that the locus of P can be expressed as |z| = 1/sqrt2
Now I've done this by turning z into Cartesian and simplifying but I was wondering why I can't divide both sides by |z+i/2| and take the reciprocal of both sides to get
|z+i|/|z+i/2| = 1/sqrt2
Since everything else in the equation is the same, I expected |z+i|/|z+i/2| to simplify into
|z|
but it didn't and I was wondering why not. Any help would be much appreciated. Thanks

$\displaystyle \frac{|z+\frac{i}{2}|}{|z+i|} = |z| \iff |z| = \frac{1}{\sqrt{2}}$

if $\displaystyle \frac{|z+\frac{i}{2}|}{|z+i|}$ simplified to give $\displaystyle |z|$ you would be saying that no matter what $\displaystyle \frac{|z+\frac{i}{2}|}{|z+i|}$ is equal to, $\displaystyle |z|$ will always equal that value which you can see clearly isnt true for z=0 ie $\displaystyle \frac{|z+\frac{i}{2}|}{|z+i|} \neq 0$ for $\displaystyle z=0$

this is a kind of special case where the solution for z when $\displaystyle \frac{|z+\frac{i}{2}|}{|z+i|} = \frac{1}{\sqrt{2}}$ is $\displaystyle |z| = \frac{1}{\sqrt{2}}$ itself


if thats hard to wrap your head around think about this instead, as its effectively the same thing;

$\displaystyle x^2 + 2x = -1 \iff x = -1$

however that does not mean that $\displaystyle x^2 + 2x = x$ for every value of x, it just so happens that $\displaystyle x^2 + 2x = -1$ gives $\displaystyle x=-1$ as a solution.

$\displaystyle x^2 + 2x = 3$ doesnt mean $\displaystyle x = 3$ just because it worked for -1

does that make sense?

The point P represents the complex number z, where
|z+i| = sqrt 2 |z+i/2| .
Show that the locus of P can be expressed as |z| = 1/sqrt2
Now I've done this by turning z into Cartesian and simplifying but I was wondering why I can't divide both sides by |z+i/2| and take the reciprocal of both sides to get
|z+i|/|z+i/2| = 1/sqrt2
Since everything else in the equation is the same, I expected |z+i|/|z+i/2| to simplify into
|z|
but it didn't and I was wondering why not. Any help would be much appreciated. Thanks

the red part is the wrong way up.