Let V be an inner product space with $x,y\in V$. Suppose $||x||^2||y||^2=|\langle x,y\rangle|^2$ . I want to show that the following holds
$$||y||^2x=\langle x,y\rangle y$$
I have that
$$||x||^2||y||^2x=|\langle x,y\rangle|^2x=\overline{\langle x, y\rangle}{\langle x, y\rangle}x$$
However this seems to put me in a loop.
I've tried to look at the properties of an inner product space, but it's doesn't seem to get me anywhere.