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

Let A and B be the line segment and points P and Q be two different mid points of AB.

So,

AP = PB

And,

AQ = QB

And,

PB + AP = AB (It coincides with line segment AB)

Similarly,

QB + AQ = AB

Now,

AP + AP = AP + BP (Since, If equals are added to equals, the wholes are equal.)

2AP = AB (i)

Similarly,

2AQ = AB (ii)

From (i) and (ii),

2AP = 2AQ (Since, things which are equal to same thing are equal to one another)

And as we know:

Things which are double of the same thing are equal to one another

Therefore,

AP = AQ

Thus, P and Q are the same points.

This contradicts the fact that P and Q are two different mid points AB.

Thus, it is proved that every line segment has one and only one mid- point.
Alternate Solution:

From the Figure,
AP + PB = AB.......................eq(i)
AQ + QB = AB.....................eq(ii)
From eq(i) and eq(ii)
AP + PB = AQ + QB
Now let AP = PB, and AQ = QB (as they are the midpoints, then)
2 AP = 2 AQ
AP = AQ
P = Q
Hence, there is only one mid-point.