Problem
GEO-B1-M01-P015 Exterior Bisector and the Base
#15
★★★☆☆ Level 3 of 5
In triangle \(ABC\), the bisector of the exterior angle at \(A\) is parallel to side \(BC\). Prove that \(AB=AC\).
Denote the angle between the exterior bisector and line \(AB\). Transfer it to the angle at \(B\).
Let the exterior bisector make an angle \(x\) with side \(AB\). Since it is parallel to \(BC\), this angle equals \(\angle B\). Since it bisects the exterior angle, the second angle between it and line \(AC\) is also \(x\). By parallelism, it equals \(\angle C\). Therefore \(\angle B=\angle C\). The triangle is isosceles, so \(AB=AC\).
This problem works well as the converse of Example 7.