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As in two other papers regarding I-cevian and I-circumcevian triangles, respectively, in this paper we propose, using logical deductibility relations and the analogy method, to present some interesting results in Triangle Geometry, more specifically regarding iterative I-circumcevian triangles. Thus, we consider a triangle ABC and the interior bisectors of the angles of the triangle, which intersect at point I and which intersect the sides of the given triangle at points A?, B? and C?, and the circumcircle of triangle ABC at A1, B1 and C1. Then, we will call the triangle A?B?C? the I-cevian triangle attached to the triangle ABC and the point I, and we will call the triangle A1B1C1 the I-circumcevian triangle attached to the triangle ABC and the point I. In the following, for any n?N, n?2, we will denote by AnBnCn the I-circumcevian triangle attached to the An-1Bn-1Cn-1 triangle. Using common mathematical knowledge, valid in any triangle, but also the results presented in the two papers mentioned above, we can obtain a series of very interesting geometric or trigonometric identities and inequalities, some of them very difficult to prove synthetically. The work is exclusively about Mathematics Didactics and is addressed to both students and teachers eager for performance in this field of Mathematics or, in Mathematics in general.