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An upper bound for a sequence of cevian inequalities. (English) Zbl 0604.51011

The altitudes, Gergonne cevians, internal angle bisectors, medians and Nagel cevians of a triangle satisfy the sequence of inequalities \(\Sigma h_ a\leq \Sigma g_ a\leq \Sigma w_ a\leq \Sigma m_ a\leq \Sigma n_ a.\) The author shows that \(\Sigma n_ a\leq 14R-19r,\) where R and r are respectively the circumradius and inradius of the triangle, with equality throughout the sequence when the triangle is equilateral.
Reviewer: P.Smith

MSC:

51M05 Euclidean geometries (general) and generalizations
52A40 Inequalities and extremum problems involving convexity in convex geometry
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