How would I go about doing this?
Background:
For $(x−y)⋅(y−z)=0$, the vectors $x−y, y−z, x−z$ create a right-angled triangle, with the hypotenuse being $x−z$. I've solved this bit: \begin{align} \|x−y+y−z\|^2 &=\langle x−y+y+z,x−y+y+z\rangle \\ &=\langle x−y,x−y>+\langle x−y,y−z\rangle +\langle y−z,z−y\rangle +\langle y−z,y−z\rangle \\ &=\|x−y\|^2+\|y−z\|^2, \end{align} but I need to prove $(x-y)⋅(y-z)=0$ using dot product, which has me stuck. So far, I have $(x−y)⋅v$, but unsure how to proceed from here. What would the next step be?