Problem
GEO-B1-M08-P009 Side Midpoints of a Quadrilateral
#9
★★★☆☆ Level 3 of 5
In convex quadrilateral \(ABCD\), points \(M,N,P,Q\) are the midpoints of sides \(AB,BC,CD,DA\). Prove that \(MNPQ\) is a parallelogram.
Draw the diagonals of the original quadrilateral.
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, using diagonal \(BD\), we get \(NP\parallel MQ\). Hence \(MNPQ\) is a parallelogram.
Mixes quadrilaterals, midlines, and an auxiliary diagonal.