Problem
GEO-B1-M04-P013 Midpoints of the Sides of a Quadrilateral
#13
★★★☆☆ Level 3 of 5
In a convex quadrilateral \(ABCD\), points \(M,N,P,Q\) are the midpoints of sides \(AB,BC,CD,DA\), respectively. Prove that \(MNPQ\) is a parallelogram.
Compare \(MN\) and \(PQ\) as midlines of triangles with diagonal \(AC\).
In triangle \(ABC\), segment \(MN\) joins the midpoints of two sides, so \(MN\parallel AC\) and \(MN=\frac{1}{2}AC\). In triangle \(CDA\), segment \(PQ\) also joins the midpoints of two sides, so \(PQ\parallel AC\) and \(PQ=\frac{1}{2}AC\). Therefore \(MN\parallel PQ\) and \(MN=PQ\). If one pair of opposite sides in a quadrilateral is both equal and parallel, then it is a parallelogram. Hence \(MNPQ\) is a parallelogram.
This is Varignon theorem in its basic form; a useful base for later problems.