Regarding orthogonality in Banach space

Question

Let $(X,\|.\|)$ be a Banach space over the real line. Let $x\neq 0$ and $y\neq 0$ be in $X$, then $x$ is said to be orthogonal to $y$ if $\|x+\lambda y\|\geq\|x\|$ for every real number $\lambda$. It is said that given any $x$ and $y$ there exists a real number $\alpha$ such that $x$ is orthogonal to $\alpha x + y$. Can anyone tell how?

For reference see Corollary 2.2 in this article. I am not able to prove the existence in the corollary, but am able to prove the rest of it.

I can see it diagrammatically, that is we need to show that $x$ is orthogonal to a point in the line passing through $y$ and in the direction of $x$. But I am not able to find this $\alpha$ rigourously

Answer 1

Consider the plane generated by $x$ and $y$. Now draw a tangent line to the sphere $\{z:\|z\|=\|x\|\}$ at point $x$. It has the form $\{x+tw|t\in \mathbb{R} \} $, and $w$ may be chosen of the form $\alpha x+y$. $x$ is orthogonal to $w$. This all works if $x$ and $y$ are not collinear; if they are the statement is formally false.