The perimeter sixpartite center of a triangle
The three medians of a triangle meet at the center of gravity and divide the triangle into six triangles of equal areas. The center of gravity is the unique area sixpartite point of a triangle. The authors of this article examine by analogy the existence of perimeter sixpartite point of a triangle. The perimeter bisecting points exists and the three corresponding cevian meet at the Nagel point of the triangle. In any triangle \(ABC\) there are three points \(A'\), \(B'\) and \(C'\) trisecting the perimeter such that the cevians \(AA'\), \(BB'\) and \(CC'\) concur at a point, called the trisected perimeter point \((X_{369})\). The perimeter bisecting deltoid of a triangle are used to study properties of three coincident perimeter bisecting lines. The most important result of the article is the following: ``Every triangle has an unique perimeter sixpartite point'' (Theorem 5). This point is non-constructible and it is not a known triangle center listed in the online Encyclopedia of triangle centers.
scientific article; zbMATH DE number 6818616