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PreprintCC BY 4.0arXiv

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

  • Tserendorj Batbold
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Abstract

Let a,b,c be the side lengths of a triangle, and let R and r denote its circumradius and inradius, respectively. Liu proposed the conjecture ∑cyc ((a(b+c−a))/bc)k ≥ 2+(2r/R)k, k>1, with the reverse inequality for k<1. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

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Cite

BibTeXRIS
BibTeX
@misc{batbold2026proof,
  title = {{A Proof of Liu's Conjecture on the Fundamental Triangle Inequality}},
  author = {Tserendorj Batbold},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10633},
  eprint = {2609.10633},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10633},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Proof of Liu's Conjecture on the Fundamental Triangle Inequality
AU  - Tserendorj Batbold
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.10633
UR  - https://arxiv.org/abs/2609.10633
AB  - Let a,b,c be the side lengths of a triangle, and let R and r denote its circumradius and inradius, respectively. Liu proposed the conjecture ∑_cyc ((a(b+c−a))/bc)^k ≥ 2+(2r/R)^k, k>1, with the reverse inequality for k<1. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.
ER  -