Using vectors to prove midpoints bisect

Question

Solved! Thank you to those who answered my question. I managed to solve it while I was away. Will edit this question with my solution so that others can check.

Problem: Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

My work:

$$U + Z = W + V$$

Let the segment from $\frac12U$ to $\frac12 W$ be vector $AB$.

Let the segment from $\frac12 Z$ to $\frac12V$ be vector $CD$.

$$\frac12(AB) = \frac12U + \frac12W + V$$

$$\frac12(CD) = \frac12Z + \frac12V + U$$

^I'm not sure if the above will help me, but it seems like the hint that was given:

Answer 1

let the vertices be $4A,4B,4C,4D$ chosen to ensure $A+B+C+D=0$ so the lines joining opposite midpoints meet at $0=\frac12(\frac12(4A+4B)+\frac12(4C+4D))$.

the midpoint of the line joining $4A$ to $4B$ is $2A+2B$, and likewise the opposite midpoint is $2C+2D$

since $0=(2A+2B)+(2C+2D)$ the origin is thus the midpoint of the either line joining the midpoints of a pair of opposite sides

Answer 2

Assuming that you can show that the midpoints are, indeed, as given in the hint, subtract the two expressions, to get

$$\frac14(u-v+z -w)=\frac14((u+z) - (v+w)) = 0,$$ the last equality being true since $u+z$ and $v+w$ are both the upper left vertex of your quadrilateral.