Problem
GEO-B1-M08-P008 A Midpoint After a Parallel
#8
★★★☆☆ Level 3 of 5
In triangle \(ABC\), point \(D\) is the midpoint of \(AB\). Through \(D\), a line parallel to \(AC\) meets \(BC\) at \(E\). Prove that \(E\) is the midpoint of \(BC\).
Look for similarity of the small and large triangles.
Since \(DE\parallel AC\), \(\triangle BDE\sim\triangle BAC\). From \(BD:BA=1:2\), we get \(BE:BC=1:2\). Hence \(BE=EC\), so \(E\) is the midpoint of \(BC\).
The method is not named: the student must recognise similarity from the parallel line.