More variations on the Steiner-Lehmus theme

Abstract

The celebrated Steiner-Lehmus theorem states that if the internal bisectors of two angles of a triangle are equal, then the triangle is isosceles. In other words, if P is the incentre of triangle ABC, and if BP and CP meet the sides AC and AB at B′and C′, respectively, then

107.11 The Steiner-Lehmus Theorem à la Ceva

“…Is Theorem 3 equivalent to some stronger form of the Steiner-Lehmus theorem? In this respect, it is worth mentioning that a stronger form of the Steiner-Lehmus theorem was indeed established in [2]. For the convenience of the reader, we state it below.…”

“…Other stronger forms of the Steiner-Lehmus theorem have appeared in the recent literature. For a most recent one, we refer the reader to [3].…”

Long medians and long angle bisectors

“…Notice that Theorem 6 (ii) is a stronger form of the classical Steiner-Lehmus theorem given by . Proofs of the stronger form (15) and other stronger forms can be found in [5,6], and other references. The proof above is adapted from that in [3].…”