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# coordinates of midpoint

The coordinates of the midpoint of a line segment are the arithmetic means of the coordinates of the endpoints of the segment. Thus, if the endpoints are $(x_{1},\,y_{1})$ and $(x_{2},\,y_{2})$, then the midpoint is

$\left(\frac{x_{1}\!+\!x_{2}}{2},\,\frac{y_{1}\!+\!y_{2}}{2}\right)\!.$ |

For justifying the above coordinates of the midpoint, we know that its abscissa $x_{0}$ halves on the $x$-axis the line segment between $x_{1}$ and $x_{2}$. Since the lengths of the half-segments are $x_{0}\!-\!x_{1}$ and $x_{2}\!-\!x_{0}$, if $x_{1}<x_{2}$, and their opposite numbers, if $x_{2}<x_{1}$, in any case we can write

$x_{0}-x_{1}=x_{2}-x_{0}.$ |

Solving this equation for $x_{0}$ yields: $\displaystyle x_{0}=\frac{x_{1}\!+\!x_{2}}{2}$. Similar result is gotten for the ordinate of the midpoint.

Related:

ConjugateDiametersOfEllipse, CentreOfMassOfPolygon, Midpoint4

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## Mathematics Subject Classification

51N20*no label found*51M15

*no label found*51-00

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