In this Geogebra sketch, a triangle is shown with a gergonne point. The gergonne point is the place where the lines from the vertices to the tangent points of the incircle intersect. The incenter point is also shown. In the sketch points A, B, and C are movable. When moving point A, B, or C the dimensions of the triangle will change along with the position of the gergonne point and the incenter.
When the triangle is an equilateral triangle the gergonne point and the incenter point are at the same possition.