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I have no idea how to proceed with this proof, any help would be greatly appreciated.

You MUST use vectors operations and/or dot product to solve these questions. Any other method is not allowed.

Take any quadrilateral. Let A, B, C and D be its vertices. Let P, Q, R and S be the middle points of its edges. Show that the quadrilateral PQRS is a parallelogram.

Hint: Show that vector SP= vector RQ and that vector SR= vector PQ

Construct line segment DB. It can be seen that triangle CBD has equal angles to triangle CQR.

This means that line segment QR is parallel to line segment DB. Using similar reasoning, it can be shown that PS is parallel to DB. Since QR and PS are both parallel to DB, they are parallel to each other.

Applying this same reasoning to line segment AC, it can be shown that PQ is parallel to SR.

Since both of the opposite sides are parallel, it is a parallelogram.

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