Triangle areas and some inferences on plane geometry - Jesus, Alessandro Firmiano de; Silva, Matheus Santos; Santos, João Paulo Martins dos (C.Q.D. - Revista Eletrônica Paulista de Matemática, 2025)
The condition for the existence of a triangle ABC with sides a, b and c establishes that a+b>c. The present work identifies in the elements of the triangle, a certain quantity k>0 such that a+b=c+k. For this identification, the intercept points M, N and P of the triangle with its incircle of center I were obtained. The main result is the Equivalence of Areas between the Re-entrant Quadrilateral ADBC, with D being the vertex running through the incircle that minimizes its area, and the area of the Deltoid (or Kite) CNIP. As a geometric contribution, a construction is suggested to obtain the intercepts M, N and P on ABC without the prior need to trace the incircle. Finally, a dynamic tool was applied to trace coordinates defined, point-to-point, by the areas of the Re-entrant Quadrilaterals ADBC, ADCB and ACDB, while the common vertex D runs through the incircle of the triangle ABC. Elliptical trajectories of these points were revealed.