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Identity
0To prove that every line segment has one and only one midpoint, we proceed in two parts:
Statement:
Every line segment has one and only one mid-point.
Part 1: Existence of a Mid-Point
Let’s consider a line segment ABABAB.
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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.
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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.
Part 2: Uniqueness of the Mid-Point
Assume, for contradiction, that two different mid-points MMM and NNN exist on segment ABABAB.
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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.
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.
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