Nikolaos Dergiades,  The Perimeter of a Cevian Triangle,

Forum Geometricorum, 2 (2002) 131--134.

Abstract:  We show that the cevian triangles of certain triangle centers have perimeters not exceeding the semiperimeter of the reference triangle. These include the incenter, the centroid, the Gergonne point, and the orthocenter when the given triangle is acute angled.

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