Problem
GEO-B1-M06-P012 Equal Areas and the Same Base
#12
★★☆☆☆ Level 2 of 5
Triangles \(ABC\) and \(ABD\) have common base \(AB\), and points \(C\) and \(D\) lie on the same side of \(AB\). It is known that \(S_{ABC}=S_{ABD}\). Prove that \(CD\parallel AB\).
Equal areas with a common base give equal heights to this base.
Since base \(AB\) is common and the areas are equal, the heights from \(C\) and \(D\) to \(AB\) are equal. Points on the same side of \(AB\) at the same distance from \(AB\) lie on a line parallel to \(AB\). Hence \(CD\parallel AB\).
A very useful converse move: deriving parallelism from equal areas.