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A geometric inequality for two interior points of the triangle. (English) Zbl 1147.26009

A geometric inequality for two interior points of the triangle is given using the basic inequality of the triangle and Stewart’s theorem. This inequality is similar to Klamkin’s inequality and the proofs uses barycentric coordinates. As corollaries some inequalities related with the medians, the altitudes, the Brocard’s point, the Gergonne’s point and the incenter of a triangle are given. These inequalities hold with equality when the respective point is one of the important points of a triangle: the barycenter, the center of gravity, the Brocard’s point, the Nagel’s point or the Gergonne’s point. An open problem related with the bisectors and the similar medians of a triangle concludes the work.

MSC:

26D15 Inequalities for sums, series and integrals
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