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Prove that the lines joining the midpoints of opposite sides of a quadrilateral and the line joining the midpoints of the diagonals of the quadrilateral are concurrent.

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Let $a$, $b$, $c$, $d$ be the position vectors of the four vertices. It is easy to check that the point $$m:={a+b+c+d\over4}$$ is the midpoint of all three mentioned line segments.

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Let $E=\frac{A+B}{2}$ and $F=\frac{C+D}{2}$ be the midpoints of two opposite sides.

Let $G=\frac{A+D}{2}$ and $H=\frac{C+B}{2}$ be the midpoints of another two opposite sides.

Let $J=\frac{A+C}{2}$ and $K=\frac{B+D}{2}$ be the midpoints of diagonals.

The midpoint of section $EF$ is a point $\frac{E+F}{2}=\frac{A+B+C+D}{2}$. Also the midpoint of $GH$ and $JK$ is $\frac{G+H}{2}=\frac{J+K}{2}=\frac{A+B+C+D}{2}$. Thuse each section $EF$, $GH$ and $JK$ contains point $L=\frac{A+B+C+D}{2}$

Therefore the lines connecting the midpoints of opposite sides of a quadrilateral and the line joining the midpoints of the diagonals of the quadrilateral are concurrent. QED

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