Problem
GEO-B1-M02-P016 The Angle Bisector Is an Altitude
#16
★★★☆☆ Level 3 of 5
In triangle \(ABC\), the angle bisector \(AD\) of angle \(A\) is perpendicular to side \(BC\). Prove that \(AB=AC\).
Compare right triangles \(ABD\) and \(ACD\).
Triangles \(ABD\) and \(ACD\) are right triangles, have common side \(AD\), and \(\angle BAD=\angle DAC\) because \(AD\) is an angle bisector. By ASA, the triangles are congruent. Therefore \(AB=AC\).
A strong basic converse: an angle-bisector altitude gives an isosceles triangle.