Question

prove that every line has one and only one mid point

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
∴ AC + BC = AB → (1)
Similarly, we get AD + DB = AB → (2)
From (1) and (2), we get
AC + BC = AD + DB
2 AC = 2AD
∴ 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 has one and only one midpoint.

hope it helped you...

Answer 2

Let us consider a line segment AB.
Assume tht it has two midpoints C and D.

so,

AC=BC and AD=BD.
Since C is the midpoint A,C and B are collinear.

AC+BC=AB   -(1) equation

AD+BD=AB   -(2)equation

 from (1) and (2) we get
AC+BC=AD+BD

2AC=2AD

ie AC=AD

This means C and D coincides.
Therefore every line segments have one and only one midpoint.