Problem
GEO-B2-M09-P005 Midline by Vectors
#5
★☆☆☆☆ Level 1 of 5
In triangle \(ABC\), points \(M,N\) are the midpoints of \(AB\) and \(AC\). Prove by vectors that \(MN\parallel BC\) and \(MN=\frac12BC\).
Write the position vectors of \(M\) and \(N\) using those of \(A,B,C\).
Let the position vectors of \(A,B,C\) be \(a,b,c\). Then \(m=\frac{a+b}{2}\), \(n=\frac{a+c}{2}\). Hence \(\overrightarrow{MN}=n-m=\frac{c-b}{2}=\frac12\overrightarrow{BC}\). This gives both parallelism and the length ratio.
A basic vector technique for later problems.