Proof for range of average We can prove that average of two numbers $a,b$ where $a<b$ will be between $a$ and $b$ as follows

$a < b$ $a + a < a + b$a <  $\dfrac{a + b}{2 }$
$a < b$ $a + b < b + b$ $\dfrac{a + b}{2 }< b$
Thus $a < \dfrac{a + b}{2 }< b$

Similarly, is it true always and can be proved, that average of three numbers, $a,b,c$ where $a<b<c$ will be between $a$ and $b$ if $b-a$ > $c-b$

Yes, it is always true.
From what you've already got, we have
$$a<b<\frac{b+c}{2}<c\implies a<\frac{b+c}{2}$$
$$\implies 2a<b+c\implies 3a<a+b+c\implies a<\frac{a+b+c}{3}$$
Also, we have
$$b-a>c-b\implies a+c\lt 2b\implies a+b+c\lt 3b\implies \frac{a+b+c}{3}\lt b$$
