Inner Product Spaces - Triangle Inequality I have to show that:
For an inner product space $V$, $\|x + y \| = \|x\| + \|y\|$, for all $x$, $y \in V$
if and only if one of the vectors $x$ or $y$ is a scalar multiple of the other.
I am thinking, if $x= cy$, for some scalar then the above equality holds. 
I am unsure how to proceed the other way.  Please help. 
Edit : I guess have to look at the proof of the triangle inequality which holds as an equality if 
$|\langle x, y\rangle | = \|x\|\,\|y\|.$

We assume that $x$ and $y$ are not $0$. If we have a relationship of the form $x=cy$ where $c$ is a scalar then $\|x\|=|c|\cdot \|y\|$ so $|c|=\frac{\|x\|}{\|y\|}$. So we try to compute $\Bigl\| \|x\|y-\|y\|x\Bigr\|^2$:
$$\begin{align*}
\Bigl\lVert \|x\|y-\|y\|x\Bigr\rVert^2&=\|x\|^2\|y\|^2+\|y\|^2\|x\|^2-\|x\|\cdot \|y\|(\langle x,y\rangle+\langle y,x\rangle)\\
&=\|x\|\cdot \|y\|(2\|x\|\cdot \|y\|-(\|x+y\|^2-\|x\|^2-\|y\|^2))\\
&=\|x\|\cdot \|y\|(2\|x\|\cdot \|y\|-2\|x\|\cdot \|y\|)\\&=0\end{align*} $$
which gives the result.
