A Mixed Altitude-Median-Exradius Inequality Associated with Two Conjectures of Jian Liu

This preprint introduces the scale-invariant mixed ratios T_a = h_a(r_b + r_c)^2/(4m_a^3) and their cyclic companions. It proves T_a^(2k) + T_b^(2k) + T_c^(2k) ≥ 1 + 2(T_a T_b T_c)^k for every nondegenerate Euclidean triangle and every real k. Equality holds exactly when k = 0 or the triangle is isosceles. Among the specified multiplicative interpolations of the altitude/median and exradius/median ratio triples, the unique all-real-exponent weight is 2/3. The motivating sources are Jian Liu, Inequalities for the product of medians in a triangle, International Journal of Geometry 14(3) (2025), 57-74, Conjectures 1 and 2 on p. 73 (https://ijgeometry.com/wp-content/uploads/2025/06/5.-57-74.pdf). Their separate proofs are already deposited at doi:10.5281/zenodo.23533433 (https://doi.org/10.5281/zenodo.23533433) and doi:10.5281/zenodo.23544282 (https://doi.org/10.5281/zenodo.23544282). The mixed theorem has its own elementary proof and does not depend on either critical endpoint certificate. A complete scalar exponent-transfer argument explains the shared analytic mechanism. Prior-result attribution. Appendix A gives an alternative proof of Liu's 2022 sum-of-powers Conjecture 5.1, posed in On the fundamental triangle inequality and Gerretsen's double inequality (https://doi.org/10.1007/s00022-022-00630-w). Batbold, arXiv:2609.10633v2, 22 September 2026, Theorem 2.2 (https://arxiv.org/abs/2609.10633v2) already proves that conjecture for all real exponents and gives equality cases. No priority for that theorem is claimed here. Methodological basis and AI assistance. The geometric methodology follows the canonical parametric framework of Canonical Parametric Representation of Plane Euclidean Geometry (https://doi.org/10.5281/zenodo.23837735) (Panorias and Theodhori, 2026), particularly normalization, algebraic formulation, verification and transfer. The arguments include both author-originated contributions and arguments developed with generative AI assistance. Initial calculations were carried out by hand and subsequently transcribed and checked with AI assistance, as stated by the author. AI also assisted verification code and text editing/typesetting. Responsibility for the final content rests with Christos Panorias. Access, rights and status. Publicly accessible preprint. Copyright (C) 2026 Christos Panorias. All rights reserved, with personal reading and private scholarly evaluation permitted as specified in the license. This work has not undergone independent peer review; its literature comparison does not certify worldwide priority.