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Some Bonnesen-style isoperimetric inequalities for triangles in the plane are presented. For example, it is shown that $L^2-12 \sqrt 3 A \geq 35.098 \,\,\, r(R-2r)$ for triangles with perimeter $L$, area $A$, inradius $r$, and circumradius $R$. Equality holds when and only when either the triangle is equilateral or the triangle is similar to the isosceles triangle with sides 1, 1, and $\lambda$ where $\lambda\approx 1.23628634$ is the largest root of the equation $31x^3-28x^2-16x+4=0$.
In this work, it is shown that a point of division divides a related line segment in the same ratio both in the taxicab and Euclidean planes. Consequently, the coordinates of the division point can be determined by the same formula as in the Euclidean plane. In the latter parts of the paper, taxicab analogues of Ceva's and Menelaus' Theorems and the theorem of directed lines are given.
The concept and some of the algebraic properties of the rejective and non-absorptive sets of a subgroup, subring, and subgroup of a module over a ring are investigated. It is shown that the set theoretic complement of a non-absorptive set in the above mentioned algebraic substructures is a normal subgroup (respectively, (left, right) ideal, submodule) of its underlying algebraic structure. The invariant property of the non-absorptive sets under the operation of inversion in the related underlying algebraic structure is proved. $G \setminus R(H)$, the set theoretic complement of the rejective set of a subgroup $H$ in a group $G$, is closed under the product in $G$ and whenever $\vert G \vert$ the order of the group $G$ is finite, $\vert R(H) \vert = (k-s) \vert H \vert$ where each of the $k$ and $s$ is the index of $H$ in $G$ and in $G \setminus R(H)$, respectively. For the case of rings and modules, the set theoretic complement of the rejective set of a substructure in the underlying ring is a subring of the underlying ring. For any subring $S$ of a ring $R$, examples and some of the properties of $S$-relative (left) ideals and $S$-relative submodules are given and also it is shown that $S$ is contained in the set theoretic complement of the rejective set of that $S$-relative (left) ideal (respectively, submodule). Finally, some of the properties of the relative homomorphisms of $R$-modules, and the rejective (respectively, non-absorptive) sets of the group homomorphisms of $R$-modules are investigated.