If a point C lies between two points A and B such that AC = BC, then 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.

Let us consider that line segment AB has two midpoints ‘C’ and ‘D’ as shown in the figure.

Let's assume C to be the mid-point of AB.

AC = BC

AC = $\frac{1}{2}$ AB-----(1)

Let us consider a point D lying on AB,

Let's assume that D is another mid-point of AB.

Therefore AD = BD

Adding equal length AD on both sides, we get

AD + AD = BD + AD (BD + AD coincides to AB)

⇒ 2 AD = AB

⇒ AD = 1/2AB------(2)

From equations (1) and (2), we can conclude that AC = AD

• C has to coincide with D for AC to be equal to AD.
• According to Euclid's axiom 4: Things which coincide with one another are equal to one another.
• Thus, a line segment has only one midpoint.