Answer
See proof
Work Step by Step
Step 1: Calculate the distance from $P_1$ to $M$.
$\sqrt {(\frac{x_1+x_2}{2}-x_{1})^{2} + (\frac{y_1+y_2}{2}-y_{1})^{2}}$
$=\sqrt {(\frac{x_2-x_1}{2})^{2} + (\frac{y_2-y_1}{2})^{2}}$
Step 2: Calculate the distance from $P_2$ to $M$.
$\sqrt {(\frac{x_1+x_2}{2}-x_{2})^{2} + (\frac{y_1+y_2}{2}-y_{2})^{2}}$
$=\sqrt {(\frac{x_2-x_1}{2})^{2} + (\frac{y_2-y_1}{2})^{2}}$
By looking at the expressions in Steps 1 and 2, we can see that the two distances are the same. Thus, $M$ is equidistant from $P_1$ and $P_2$, confirming that $M$ is indeed the midpoint of $P_1$ and $P_2$.
