Problem
GEO-B2-M09-P010 Varignon Parallelogram
#10
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In an arbitrary quadrilateral \(ABCD\), points \(P,Q,R,S\) are the midpoints of \(AB,BC,CD,DA\). Prove that \(PQRS\) is a parallelogram.
Write \(\overrightarrow{PQ}\) and \(\overrightarrow{SR}\) using the position vectors of the vertices.
Let the position vectors of \(A,B,C,D\) be \(a,b,c,d\). Then \(p=\frac{a+b}{2}\), \(q=\frac{b+c}{2}\), \(r=\frac{c+d}{2}\), \(s=\frac{d+a}{2}\). We have \(\overrightarrow{PQ}=q-p=\frac{c-a}{2}\) and \(\overrightarrow{SR}=r-s=\frac{c-a}{2}\). Similarly, \(\overrightarrow{QR}=\overrightarrow{PS}\). Hence \(PQRS\) is a parallelogram.
A classical example where vectors leave almost no work.