Problem
GEO-B1-M07-P015 An Inner Parallelogram
#15
★★★☆☆ Level 3 of 5
In triangle \(ABC\), point \(D\) lies on side \(BC\). Through \(D\), lines parallel to \(AB\) and \(AC\) are drawn; they meet \(AC\) and \(AB\) at points \(E\) and \(F\), respectively. Prove that \(AEDF\) is a parallelogram.
Check which sides of quadrilateral \(AEDF\) are parallel.
Since \(E\) lies on \(AC\), side \(AE\) is parallel to the line through \(D\) parallel to \(AC\), namely \(DF\). Since \(F\) lies on \(AB\), side \(AF\) is parallel to the line through \(D\) parallel to \(AB\), namely \(DE\). Thus \(AE\parallel DF\) and \(AF\parallel DE\), so \(AEDF\) is a parallelogram.
A standard way to add a parallelogram inside a triangle.