If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$,then prove that $AC = \frac{1}{2} AB$. Point $C$ is called a mid-point of line segment $AB$. Prove that every line segment has one and only one mid-point.

(N/A) Given that $C$ is a mid-point of line segment $AB$,so $AC = BC$.
Since $AC + BC = AB$,we can write $AC + AC = AB$,which implies $2AC = AB$,or $AC = \frac{1}{2} AB$.
To prove that every line segment has one and only one mid-point,let us assume that there are two distinct mid-points $C$ and $D$ for the line segment $AB$.
Since $C$ is a mid-point,$AC = \frac{1}{2} AB$ ............. $(1)$
Since $D$ is a mid-point,$AD = \frac{1}{2} AB$ ............. $(2)$
From $(1)$ and $(2)$,we have $AC = AD$.
This implies that points $C$ and $D$ must coincide.
Thus,every line segment has one and only one mid-point.