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
Recall the vector \(\vec {EF}\) is given by
\(\vec {EF}\) = position vector of F - position vector of E
Similarly \(\vec{HG}\) = position vector of G - position vector of H
So, we have \(\vec{EF}=\vec{HG}\).
Two vectors are equal only when both their magnitudes and directions are equal.
⇒ \(\vec {EF}||\vec{HG}\) and \(|\vec{EF}|=|\vec {HG}|.\)
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.
