Arnab Bhattacharjee

To prove that every line segment has one and only one midpoint, we proceed in two parts:

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Statement:

Every line segment has one and only one mid-point.

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Part 1: Existence of a Mid-Point

Let’s consider a line segment ABABAB.

• A mid-point is a point MMM on the segment such that:

  AM=MBAM = MBAM=MB

• According to geometry (Euclidean postulates), we can construct a point on segment ABABAB such that the distance from AAA to MMM is equal to the distance from MMM to BBB.

• In coordinate geometry, if A(x1,y1)A(x_1, y_1)A(x1,y1) and B(x2,y2)B(x_2, y_2)B(x2,y2), then the midpoint MMM has coordinates:

  M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)M=(2x1+x2,2y1+y2)

This guarantees that a mid-point always exists.

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Part 2: Uniqueness of the Mid-Point

Assume, for contradiction, that two different mid-points MMM and NNN exist on segment ABABAB.

• Then both must satisfy:

  AM=MBandAN=NBAM = MB \quad \text{and} \quad AN = NBAM=MBandAN=NB

• But in a straight line segment, there can only be one point that is equidistant from both AAA and BBB.
  If M≠NM \ne NM=N, then either AM≠ANAM \ne ANAM=AN or MB≠NBMB \ne NBMB=NB, which contradicts the definition of a midpoint.

Hence, no two distinct points on a segment can both be midpoints.

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Conclusion:

Every line segment has one and only one midpoint — it exists due to geometric construction, and it's unique because no other point can fulfill the same condition.