Problem
GEO-B1-M07-P011 Midline Through a Construction
#11
★★☆☆☆ Level 2 of 5
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of \(AB\) and \(AC\). Prove that \(MN\parallel BC\), using an auxiliary point on line \(MN\).
Extend \(MN\) beyond \(N\) to point \(P\) so that \(NP=MN\).
Construct point \(P\) on the extension of \(MN\) so that \(NP=MN\). In triangles \(AMN\) and \(CNP\), we have \(AN=NC\), \(MN=NP\), and the angles at \(N\) are vertical. Hence the triangles are congruent, so \(AM=CP\) and \(AM\parallel CP\). Since \(AM=MB\), we get \(CP=MB\) and \(CP\parallel MB\), so \(BMCP\) is a parallelogram. Therefore \(MP\parallel BC\), hence \(MN\parallel BC\).
This is a constructive proof of the midline theorem.