Let ABCD be a quadrilateral. E, F, G and H are the midpoints of sides AB, BC, CD and DA respectively.
We need to prove EG and HF bisect each other. It is sufficient to show EFGH is a parallelogram, as the diagonals in a parallelogram bisect each other.
Let the position vectors of these vertices and midpoints be as shown in the figure.
As E is the midpoint of AB, using midpoint formula, we have
Similarly, , and .
Recall the vector is given by
So, we have .
Two vectors are equal only when both their magnitudes and directions are equal.
This means that the opposite sides in quadrilateral EFGH are parallel and equal, making EFGH a parallelogram.
EG and HF are diagonals of parallelogram EFGH. So, EG and HF bisect each other.
Thus, the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.
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