Prove that conditions are equivalent

Question

$ X $ is unitary space, $ x,y \in X $. Prove that following conditions are equivalent:

$ x \perp y $

$ ||x|| \leq ||x+ty|| $ $ t \in C $

$ ||x+ty||=||x-ty|| $ $ t \in C $

Unfortunatelly, I'm not able to solve it

I was thinking about using triangle inequality

$ ||x|| = ||x+ty-ty|| \leq .. $

Thank you in advance

Answer 1

Use the fact that the norm is induced by an inner product, then use the properties of said inner product, e.g. $$ \| x+ty\|^2 = \langle x+ty, x+ty\rangle = \|x\|^2 + t^2 \|y\|^2 + 2t\langle x,y \rangle $$ and then use the fact that $x$ and $y$ are orthogonal.