Existence of the Gergonne point. Let L, M, and N be the points at which the incircle touches the sides of triangle ABC. Prove that AL, BM, and CN are concurrent. Hint: Use the external tangents theorem.
Triangle ABC has an incircle with Incenter denoted I. The incircle is tangent to the sides of the triangle at points D, E, and F. The Gergonne Point is the point of concurrency of the segments AD, BE, and CF. The proof of concurrency is needed. Observe that the Incenter lies on the anglebisectors of the triangle and the radii of the incircle, ID, IE, and IF are perpendicular to the sides. Thus, three pairs of congruent right triangles are formed. The products (AF)(BD)(CE) and (AE)(CD)(BF) are equal because BF = BD, CD = CE, and AE = AF. Therefore, the ratio of the products is 1 and by Ceva’s Theorem the three Cevians of the triangle (AD, BE, and CF) are concurrent.