One group of inequalities with altitudes and medians in triangle

In the article we prove some inequalities that contain relations between altitudes and medians in triangle. At least one of these inequalities has not been considered in the literature before and the main theorem has also not been proved elsewhere in that form. Some immediate corollaries have been presented as well.

💡 Research Summary

The paper investigates a family of inequalities that link the altitudes and the medians of a triangle. After a brief review of known results, the authors introduce several new relations, prove them rigorously, and derive a number of corollaries.

Notation and preliminaries.
For a triangle (ABC) with side lengths (a,b,c), semiperimeter (s=\frac{a+b+c}{2}), area (\Delta), circumradius (R) and inradius (r), the altitudes are written as
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