Seemingly easy Question concerning parametric lines: if $z= 0.5x + 0.5y $ , show that $\|x-z\|=\|y-z\|$ .

Question

The parametric equation I am working with assumes that two points $x,y$ lie on the line, which can be understood as going in the direction $y-x$, thus the equation will be, $x(t)= (1-t)x + ty $ For $t= 0.5$ I get the equation in the question. But am no closer to getting an answer. So please help me out.

Answer 1

Let $p(t) = ty+(1-t)x = x + t(y-x)$.

Note that $p(t)-p(s) = (t-s)(y-x)$.

Then $z-x = {1 \over 2}(y-x)$ and $y-z = {1 \over 2}(y-x)$.