if $Re(z\overline w)=|z||w|$ then $w=tz$ where $t\in \mathbb R$ I was given the following exercise in complex numbers:
Assume that for some $z,w \in \mathbb C$: $Re(z\overline  w)=|z||w|$
Show that this implies $w=tz$ where $t$ is some positive real.
What I did: we know that
$$Re(z\overline  w)=\frac{z\overline w+\overline z w}{2}=|z||w|$$
Then $$Re(z\overline  w)^2=(\frac{z\overline w+\overline z w}{2})^2=\frac{(z\overline w+\overline z w)(z\overline w+\overline z w)}{4}=\frac{z^2\overline w^2+2z\overline z w\overline w+\overline z^2w^2}{4}=|z|^2|w|^2=z \overline z w\overline w$$
Then overall we see that $z^2\overline w^2+\overline z^2w^2=2z\overline z w\overline w$
I don't see how this implies that $w=tz$. I don't see the connection.
What am I missing?

You have
$$\frac{\overline zw+z\overline w}{2}=|z||w|$$
Dividing both sides by $z\overline z=|z|^2$, we get:
$$\frac12\left(\frac wz+\overline{\frac wz}\right)=\frac{|w|}{|z|}$$
or:
$$\operatorname{Re}\left(\frac wz\right)=\left|\frac wz\right|$$
If $\operatorname{Re}(x)=|x|$, then we know that $x$ is real. Thus, $\frac wz$ is real.
