Introduction to Euclid's Geometry — Question 5

Let us consider that line segment AB has two midpoints C and D.

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

∴ AC = BC

Adding AC on both sides, we get :

⇒ AC + AC = BC + AC

⇒ 2AC = AB

⇒ AC = $\dfrac{1}{2}AB$ .............(1)

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

∴ AD = BD

Adding equal length AD on both sides, we get

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

⇒ 2AD = AB

⇒ AD = $\dfrac{1}{2}AB$ .........(2)

From equations (1) and (2), we get :

⇒ 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.

Hence, proved that a line segment has only one midpoint.