An elementary proof of the Liu conjecture on triangle altitudes and medians

The first conjecture of Liu concerns a parameter-dependent inequality involving the ratios of the altitudes of a triangle to the corresponding medians. An elementary proof is given for the full conjectured range k ≥ −3. The argument begins with a common translation and rotation of the triangle. In the resulting coordinates, each height-to-median ratio is determined by a difference of squared side lengths divided by four times the area. These three quantities sum to zero, reducing the problem to two nonnegative parameters. Two polynomial identities establish the endpoint k = −3. Concavity then extends the result to the remaining negative exponents, while a trigonometric argument treats the positive exponents. The equality cases are determined, and triangles approaching the equilateral triangle show that the lower endpoint cannot be decreased. 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.