Curves Related to the Gergonne Point in an Isotropic Plane

The notion of the Gergonne point of a triangle in the Euclidean plane is very well known, and the study of them in the isotropic setting has already appeared earlier. In this paper, we give two generalizations of the Gergonne point of a triangle in the isotropic plane, and we study several curves related to them. The first generalization is based on the fact that for the triangle A B C and its contact triangle A i B i C i , there is a pencil of circles such that each circle k m from the pencil the lines A A m , B B m , C C m is concurrent at a point G m , where A m , B m , C m are points on k m parallel to A i , B i , C i , respectively. To introduce the second generalization of the Gergonne point, we prove that for the triangle A B C , point I and three lines q 1 , q 2 , q 3 through I there are two points G 1 , 2 such that for the points Q 1 , Q 2 , Q 3 on q 1 , q 2 , q 3 with d ( I , Q 1 ) = d ( I , Q 2 ) = d ( I , Q 3 ) , the lines A Q 1 , B Q 2 and C Q 3 are concurrent at G 1 , 2 . We achieve these results by using the standardization of the triangle in the isotropic plane and simple analytical method.