Problem
GEO-B1-M07-P021 Midpoints of a Quadrilateral
#21
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In convex quadrilateral \(ABCD\), points \(M,N,P,Q\) are the midpoints of sides \(AB,BC,CD,DA\). Prove that \(MNPQ\) is a parallelogram by choosing the right auxiliary lines.
Draw diagonals \(AC\) and \(BD\).
Draw \(AC\). In triangle \(ABC\), segment \(MN\) is a midline, so \(MN\parallel AC\). In triangle \(CDA\), segment \(PQ\) is a midline, so \(PQ\parallel AC\). Therefore \(MN\parallel PQ\). Similarly, drawing \(BD\), we get \(NP\parallel MQ\). Hence \(MNPQ\) is a parallelogram.
A diagonal is one of the most natural auxiliary lines in a quadrilateral.