point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point

5. In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Let C and D are two midpoints of the line segments AB

According to Euclid’s axioms 4

AC     = BC           ...(1)

D is also a mid point so that

AD     = DB           … (2)

We have                AB     = AB            … (3)

And  we know       AB     = AC + CB  

AB     = AD + DB

From equation (iii)

AC + CB      = AD + DB

From equation (1) and (2)plug the value of BC and DB we get

AC +AC       = AD +AD

2AC   = 2AD

Divide by 2 we get

AC     = AD

Both points are on same line so both points will superimpose and D and C are exactly at the same place

Hence midpoint of the lines segment is always unique