Problem
GEO-B1-M01-P019 A Bisector After Drawing Parallels
#19
★★★☆☆ Level 3 of 5
In triangle \(ABC\), point \(D\) lies on side \(BC\). Through \(D\), draw lines \(DE\parallel AB\) and \(DF\parallel AC\), where \(E\in AC\), \(F\in AB\). Prove that if \(AD\) bisects angle \(A\), then \(AD\) bisects angle \(EDF\).
Transfer angles \(BAD\) and \(DAC\) to point \(D\) using the parallel lines.
Since \(DE\parallel AB\), the angle between \(DE\) and \(AD\) equals \(\angle BAD\). Since \(DF\parallel AC\), the angle between \(AD\) and \(DF\) equals \(\angle DAC\). By the condition, \(\angle BAD=\angle DAC\). Hence \(AD\) divides \(\angle EDF\) into two equal angles, so it is its angle bisector.
This is the first problem with an explicit auxiliary parallel construction.