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A Wild Steiner-Lehmus Chase

Eric L. Grinberg, Mehmet Z. Orhon

TL;DR

This paper provides an elementary geometric reproof of the Steiner-Lehmus Theorem by comparing the A- and B-angle bisectors via successive applications of the Law of Sines. The core argument introduces an auxiliary segment and an intermediate point, uses a reflection for visualization, and relies on the monotonicity of the sine function to show that the larger angle yields a shorter bisector, i.e., if $\angle A > \angle B$ then the A-bisector length $\ell$ is less than the B-bisector length $\ell'$. The authors frame the proof within a broader discussion of indirect versus direct proofs, constructive versus non-constructive methods, and the historical practice of reproofs, including the idea that such proofs can inspire future direct approaches. They also acknowledge the visualization value of the reflection step and situate their result in a lineage of related proofs, highlighting both the mathematical and pedagogical significance of reproof culture.

Abstract

We present a proof the Steiner-Lehmus equal bisectors theorem by applying the Law of sines in rapid succession to a side-by-side comparison. For nearly two centuries, the quest for a direct proof has sustained interest in proving and reproving this theorem. We suggest that a second driving force may also be at play.

A Wild Steiner-Lehmus Chase

TL;DR

This paper provides an elementary geometric reproof of the Steiner-Lehmus Theorem by comparing the A- and B-angle bisectors via successive applications of the Law of Sines. The core argument introduces an auxiliary segment and an intermediate point, uses a reflection for visualization, and relies on the monotonicity of the sine function to show that the larger angle yields a shorter bisector, i.e., if then the A-bisector length is less than the B-bisector length . The authors frame the proof within a broader discussion of indirect versus direct proofs, constructive versus non-constructive methods, and the historical practice of reproofs, including the idea that such proofs can inspire future direct approaches. They also acknowledge the visualization value of the reflection step and situate their result in a lineage of related proofs, highlighting both the mathematical and pedagogical significance of reproof culture.

Abstract

We present a proof the Steiner-Lehmus equal bisectors theorem by applying the Law of sines in rapid succession to a side-by-side comparison. For nearly two centuries, the quest for a direct proof has sustained interest in proving and reproving this theorem. We suggest that a second driving force may also be at play.
Paper Structure (7 sections, 1 theorem, 5 equations, 4 figures)

This paper contains 7 sections, 1 theorem, 5 equations, 4 figures.

Table of Contents

  1. Every Proof Tells a Story
  2. Theorems That Keep On Giving.
  3. Proving The Steiner-Lehmus Theorem.
  4. The Oldest Indirect Proof?
  5. Constructive Criticism and Indirect Costs
  6. Dèjá Vu All Over Again?
  7. Why Reprove it?

Key Result

Theorem 1

Let the triangle $T \equiv \triangle ABC$ have base angles $\angle A, \angle B$, with $\angle A$ larger than $\angle B$. Then the internal bisecting segment of $\angle A$ in $T$ is shorter than the internal bisecting segment of $\angle B$ in $T$.

Figures (4)

  • Figure 1: Triangle with first bisector
  • Figure 2: Triangle with second bisector
  • Figure 3: Triangles, upon reflection
  • Figure :

Theorems & Definitions (1)

  • Theorem