Problem
GEO-B1-M07-P022 Trapezoid Midline Through a Diagonal
#22
★★★★☆ Level 4 of 5
In trapezoid \(ABCD\), bases \(AD\parallel BC\). Points \(M\) and \(N\) are the midpoints of legs \(AB\) and \(CD\). Prove that \(MN\parallel AD\), by drawing an auxiliary diagonal.
Draw diagonal \(AC\) and mark its midpoint.
Draw \(AC\), and let \(P\) be its midpoint. In triangle \(ABC\), segment \(MP\) is a midline, so \(MP\parallel BC\). In triangle \(ACD\), segment \(PN\) is a midline, so \(PN\parallel AD\). Since \(AD\parallel BC\), lines \(MP\) and \(PN\) coincide. Thus \(M,P,N\) are collinear, and \(MN\parallel AD\).
Shows how a diagonal and an added midpoint turn a trapezoid into two midline problems.