Question

Prove that every line segment has a unique mid-point.

Answer 1

Let us assume that a line segment can have two mid-points, C and D, and try to prove this.
Let C and D be the two mid-points of line segment AB.
According to Euclid's fourth axiom,
AC = BC .....(1)
D is also a mid-point. So,
AD = DB ....(2)
We have: AB = AB .....(3)
And we know that AB = AC +CB.
AB = AD +DB
From equation (3), we get:
​AC +CB = ​AD +DB
Substituting the values of BC and DB from equations (1) and (2), in equation (3), we get:
AC + AC = AD + AD
2AC = 2AD
Dividing both sides by 2, we get:
AC = AD
This is a contradiction, unless C and D coincide.
Therefore ,our assumption that a line segment AB has two mid-points is incorrect.
Thus, every line segment has a unique mid-point.
Hence proved.