Problem
GEO-B2-M04-P023 Two Parallel Sections of a Triangle
In triangle \(ABC\), two segments \(D_1E_1\) and \(D_2E_2\), both parallel to \(BC\), are drawn, where \(D_1,D_2\) lie on \(AB\), and \(E_1,E_2\) lie on \(AC\). Let \(M_1\) and \(M_2\) be the midpoints of \(D_1E_1\) and \(D_2E_2\), and let \(M\) be the midpoint of \(BC\). Prove that points \(A,M_1,M_2,M\) are collinear.
Each parallel section is the image of \(BC\) under a homothety with centre \(A\).
Since \(D_iE_i \parallel BC\), segment \(D_iE_i\) is the image of \(BC\) under a homothety with centre \(A\) and some ratio \(k_i\). Under this homothety, the midpoint \(M\) of \(BC\) is sent to the midpoint \(M_i\) of \(D_iE_i\). Therefore points \(A,M,M_i\) are collinear for \(i=1,2\). Hence \(A,M_1,M_2,M\) are collinear.
A good problem on transporting the midpoint property through a family of homotheties.