Problem
GEO-B2-M04-P011 Midpoints and One Line
#11
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D\) and \(E\) are the midpoints of sides \(AB\) and \(AC\), and \(M\) is the midpoint of \(BC\). Prove that the midpoint of segment \(DE\) lies on line \(AM\).
Consider the homothety with centre \(A\) and ratio \(\frac{1}{2}\).
The homothety with centre \(A\) and ratio \(\frac{1}{2}\) sends \(B\) to \(D\), \(C\) to \(E\), and segment \(BC\) to \(DE\). A midpoint is sent to the midpoint of the image. Therefore the midpoint \(M\) of \(BC\) is sent to the midpoint \(N\) of \(DE\). A point, its image, and the centre of homothety are collinear, so \(A,M,N\) are collinear.
The problem shows that homothety works not only with vertices but with all points of a segment.