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Journal for Geometry and Graphics 14 (2010), No. 2, 127--133
Copyright Heldermann Verlag 2010

More on the Steiner-Lehmus Theorem

Sadi Abu-Saymeh
Dept. of Mathematics, Yarmouk University, Irbid, Jordan

Mowaffaq Hajja
Dept. of Mathematics, Yarmouk University, Irbid, Jordan

For any point P in the plane of the triangle ABC, we let BBP, CCP be the cevians through P. Then the Steiner-Lehmus Theorem states that if I is the incenter of ABC and if BBI = CCI 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 BBP = CCP implies AB = AC, or equivalently whether AB ≠ AC implies BBP ≠ CCP. For a triangle ABC with AB ≠ AC, we describe a line segment XY on the line AJ inside of which there exists P with BBP = CCP and ouside of which there are no such points.

Keywords: Steiner-Lehmus theorem, cevian, Ceva's theorem.

MSC: 51M04

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