
Journal for Geometry and Graphics 14 (2010), No. 2, 127133 Copyright Heldermann Verlag 2010 More on the SteinerLehmus Theorem Sadi AbuSaymeh Dept. of Mathematics, Yarmouk University, Irbid, Jordan sade@yu.edu.jo Mowaffaq Hajja Dept. of Mathematics, Yarmouk University, Irbid, Jordan mhajja@yu.edu.jo For any point P in the plane of the triangle ABC, we let BB_{P}, CC_{P} be the cevians through P. Then the SteinerLehmus Theorem states that if I is the incenter of ABC and if BB_{I} = CC_{I} then AB = AC. Letting the internal angle bisector of A meet BC at J, it is stated by V. Nicula and C. Pohoata that the same holds if I is replaced by any point on the ray AJ. However, the proof there is valid for points on segment AJ and for points on the extension of AJ that are not very far away from side BC. In this paper, we consider all points P on the line AJ and we answer the question whether BB_{P} = CC_{P} implies AB = AC, or equivalently whether AB ≠ AC implies BB_{P} ≠ CC_{P}. For a triangle ABC with AB ≠ AC, we describe a line segment XY on the line AJ inside of which there exists P with BB_{P} = CC_{P} and ouside of which there are no such points. Keywords: SteinerLehmus theorem, cevian, Ceva's theorem. MSC: 51M04 [ Fulltextpdf (113 KB)] for subscribers only. 