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If A,B,C,D are any four points in the 3D space, and K,L,M,N are the middles of line segments AB,BC,CD,DA, then KLMN is a parallelogram. How can I prove this using vectors?

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Start by sketching it in 2-d space e.g. on a piece of paper to persuade yourself it's true.

$$\vec{KL} = \vec{KD} + \vec{DL} = \frac{1}{2} \left(\vec{AD} + \vec{DC} \right) = \frac{1}{2} \vec{AC}$$

Similarly for $\vec{NM}$.

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