A generalized Seebach’s theorem

Abstract

We present a generalized version of Seebach’s theorem considering the shapes of Cevian triangles with respect to points not only inside but also outside the reference triangle \(ABC\). The generalized version claims that for each positively oriented triangle \(A_1B_1C_1\) we have one to three points in the plane with positively oriented Cevian triangles directly similar to \(A_1B_1C_1\). With one exception the same is true for negatively oriented Cevian triangles. An information on trilinear coordinates of these points is provided. As an application we consider points having Cevian triangles directly similar to the reference triangle. In the positively oriented case besides the obvious candidate, the centroid, we found two additional such points in obtuse triangles: the intersections of the de Longchamps line with the circumcircle. As another application we consider points with equilateral Cevian triangles and present an example of a triangle in which the Kimberling triangle center \(X_{370}\) is not constructible by ruler and compass.