midpoint(a + b) = midpoint(a) + midpoint(b) - Help understanding

Question

Consider two real numbers $a=2$ and $b=6$.

Goal is to obtain their midpoint, so in order to use symmetry I add $a$ to the other side of $b$ and can see easily from the below figure: midpoint of $a$, $b$ = midpoint of $b+a$.

Good so far. What I'm stuck at is interpreting the quantities $\frac{a}{2}$ and $\frac{b}{2}$ geometrically. Visually is there a nice way to convince $\frac{a}{2} + \frac{b}{2}$ is the midpoint of $a,b$ ?

Answer 1

What you are really showing is midpoint of $(a+b)$ and $0$ is midpoint of $a$ and $0$ plus midpoint of $b$ and $0$ that is

$$ \frac {(a+b)+0}{2} = \frac {a+0}{2} + \frac {b+0}{2}$$

In general we do not have midpoint of $(a+b)$ and $c$ is the same as midpoint of $a$ and $c$ plus midpoint of $b$ and $c$, that is in general $$ \frac {(a+b)+c}{2}\ne \frac {a+c}{2} + \frac {b+c}{2}$$

Answer 2

It is $$\frac{a+b}{2}=\frac{a}{2}+\frac{b}{2}$$

Answer 3

Here the geometric interpretation you mean:

You can see firstly $\frac{a}{2}$ and secondly $\frac{b}{2}$.