Gergonne point

The Gergonne point is a special point within a triangle, found by connecting each vertex to the point where the triangle's incircle (the circle inside touching all sides) touches the sides. When these three lines—each from a vertex to its respective tangent point—intersect, they do so at the Gergonne point. This point has interesting properties related to the triangle's internal division and can be used in advanced geometrical constructions and proofs, highlighting the elegant relationships between a triangle’s incircle and its vertices.