Nesrine Benyahia Tani
Zahra Yahi
Abstract:   Using several arguments, some authors showed that the number of non-congruent triangles inscribed in a regular \mbox{$n$-gon} equals $\{n^{2}/12\}$, where $\left\{ x\right\}$ is the nearest integer to $x$. In this paper, we revisit the same problem, but study the number of ordered and non-ordered non-congruent convex quadrilaterals, for which we give simple closed formulas using Partition Theory. The paper is complemented by a study of two further kinds of quadrilaterals called proper and improper non-congruent convex quadrilaterals, which allows to give a formula that connects the number of triangles and ordered quadrilaterals. This formula can be considered as a new combinatorial interpretation of a certain identity in Partition Theory.