Coordinate Proofs Math Example 2

Solution

1. 1
   Step 1: Let the quadrilateral have vertices $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, $D(x_4,y_4)$ in order.
2. 2
   Step 2: Find the midpoints of each side: $M_{AB} = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$, $M_{BC} = \left(\frac{x_2+x_3}{2}, \frac{y_2+y_3}{2}\right)$, $M_{CD} = \left(\frac{x_3+x_4}{2}, \frac{y_3+y_4}{2}\right)$, $M_{DA} = \left(\frac{x_4+x_1}{2}, \frac{y_4+y_1}{2}\right)$.
3. 3
   Step 3: Compute the midpoint of diagonal $M_{AB}M_{CD}$: $\left(\frac{\frac{x_1+x_2}{2}+\frac{x_3+x_4}{2}}{2}, \frac{\frac{y_1+y_2}{2}+\frac{y_3+y_4}{2}}{2}\right) = \left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}\right)$.
4. 4
   Step 4: Compute the midpoint of diagonal $M_{BC}M_{DA}$: the same result $\left(\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}\right)$.
5. 5
   Step 5: Since the diagonals of quadrilateral $M_{AB}M_{BC}M_{CD}M_{DA}$ bisect each other, it is a parallelogram. $\blacksquare$