Question

Prove that every line segment has a unique midpoint.

Answer 1

Let us consider, a line segment AB. Assume that it has two midpoints say C and D
Recall that the midpoint of a line segment divides it into two equal parts
That is AC = BC and AD = DB
Since C is midpoint of AB, we have A, C and B are collinear
$\therefore AC+BC=AB\to$ (1)
Similarly, we get AD + DB = AB $\to$ (2)
From (1) and (2), we get
AC + BC = AD + DB
$\Rightarrow$ 2 AC = 2 AD
$\therefore$ AC = AD
This is a contradiction unless C and D coincide.Therefore our assumption that a line segment AB has two midpoints is incorrect.Thus every line segment will have only one unique mid point midpoint.