Question

Show that every line segment has one and only one middle point.

Answer 1

Let us consider a line segment AB.
Assume that it has two mid-points, say C and D.

Recall that the midpoint of a line segment divides it into two equal parts and both C and D are mid-points.
That is, AC = BC and AD = BD.
Since C is the midpoint of AB, A, C and B are collinear.
Therefore, AC + BC = AB..............(1)
Similarly, AD + DB = AB ......(2)
From (1) and (2), we get:
AC + BC = AD + DB
Or, 2AC = 2 AD
Therefore, 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 one and only one middle point.
Hence proved.