A computer-assisted proof of Liu's conjecture on triangle medians and exradii

A proof of Conjecture 2 in Jian Liu's 2025 article Inequalities for the product of medians in a triangle is presented for the full range k >= -12. Ravi variables yield rational formulas for the squared median-exradius ratios. At k = -12 a symmetric homogeneous polynomial of degree 126 factors, in ordered difference variables, into a squared difference factor times a degree-120 polynomial with 6165 strictly positive nonzero integer coefficients. The complete finite certificate, two different exact verification implementations, a compact self-contained endpoint verifier and universal symbolic geometry checks accompany the manuscript. Increasing convex transformations of cosine and hyperbolic-cosine coordinates establish the remaining exponents. All equality cases are determined, and a family of actual acute triangles proves sharpness of the exponent -12. This is a preprint and has not undergone independent peer review. The PDF and the supplementary ZIP together provide the proof and its reproducibility materials. Use of AI tools. ChatGPT was used solely for language editing. All mathematical ideas, arguments, calculations, implementations and code are the authors' own work. The authors are responsible for the content.