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 ABC and its contact triangle AiBiCi, there is a pencil of circles such that each circle km from the pencil the lines AAm, BBm, CCm is concurrent at a point Gm, where Am, Bm, Cm are points on km parallel to Ai,Bi,Ci, respectively. To introduce the second generalization of the Gergonne point, we prove that for the triangle ABC, point I and three lines q1,q2,q3 through I there are two points G1,2 such that for the points Q1,Q2,Q3 on q1,q2,q3 with d(I,Q1)=d(I,Q2)=d(I,Q3), the lines AQ1,BQ2 and CQ3 are concurrent at G1,2. We achieve these results by using the standardization of the triangle in the isotropic plane and simple analytical method.