For simplicity, the points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ will be arbitrarily plotted in Quadrant I. Also, consider the line segment that connects these points. The midpoint $M$ between $A$ and $B$ is the midpoint of this segment. Note that the position of the points in the plane does not affect the proof. Consider the horizontal distance $\Delta x$ and the vertical distance $\Delta y$ between $A$ and $B$. Since $M$ is the midpoint, $M$ splits each distance, $\Delta x$ and $\Delta y$, in half. Therefore, the horizontal and vertical distances from each endpoint to the midpoint are $\frac{\Delta x}{2}$ and $\frac{\Delta y}{2}.$ Let $x_m$ and $y_m$ be the coordinates of $M.$ Now, focus on the $x\text{-}$coordinates. The difference between the corresponding $x\text{-}$coordinates gives the horizontal distances between the midpoint and the endpoints. The graph above shows that these distances are both equal to $\frac{\Delta x}{2}.$ Therefore, by the Transitive Property of Equality, they are equal. This equation can be solved to find $x_m,$ the $x\text{-}$coordinate of the midpoint $M.$ The $x\text{-}$coordinate of $M$ is $x_m=\frac{x_1+x_2}{2}.$ In the same way, it can be shown that the $y\text{-}$coordinate of $M$ is $y_m=\frac{y_1+y_2}{2}.$ With this information, the coordinates of $M$ can be expressed in terms of the coordinates of $A$ and $B.$