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

$$(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}).$$

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 ,  and their opposite numbers, if  ,  in any case we can write

$$x_0-x_1=x_2-x_0.$$

Solving this equation for $x_0$ yields:  $x_0={\displaystyle \frac{x_1+x_2}{2}}$.  result is gotten for the ordinate of the midpoint.