In a normed space and given three points $x,y,z $ the norm satisfies the triangle inequality
$$\|x-y \| \le \|x-z \| + \|z-y \|$$
But is also the intuitively obvious fact from $\mathbb R^2$ true that if $z$ is a point on the line $x+t(x-y)$, $0 < t < 1 $, between $x $ and $y $ then the distances betewwen $x $ and $z $ , and $z $ and $y $ add up to the distance between $x $ and $y $
? And how would we prove it?
Most grateful for any help!