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Congruent triangles in arrangements of lines. (English) Zbl 1396.52022
In the paper under review, the author studies the maximal number of congruent triangles determined by finite arrangements of $$\ell$$ lines in $$\mathbb{R}^{2}$$. In order to formulate the main results of the paper, we need to recall some notions. An arrangement $$\mathcal{A}$$ of lines is always a finite family of $$\ell$$ lines $$L_{1},\dots,L_{\ell}$$, which is not a pencil. Denote by $$\mathfrak{U}_{\ell}$$ the set of all arrangements of $$\ell \geq 3$$ lines. For an arrangement $$\mathcal{A} \in \mathfrak{U}_{\ell}$$ we can associate a graph $$\Gamma_{\mathcal{A}}$$: the vertices of $$\Gamma_{\mathcal{A}}$$ correspond to the intersection points of lines and the edges of $$\Gamma_{\mathcal{A}}$$ correspond to the line-segments between these vertices. A triangle in $$\mathcal{A} \in \mathfrak{U}_{\ell}$$ is the convex hull of the set of intersection points of three non-concurrent pairwise non-parallel lines in $$\mathcal{A}$$. We denote by $$F^{\mathcal{A}}$$ the set of all triangles in $$\mathcal{A}$$. If two triangles $$\triangle_{1}, \triangle_{2} \in F^{\mathcal{A}}$$ are congruent, then we write $$\triangle_{1} \sim \triangle_{2}$$. Let $$F_{1}^{\mathcal{A}}, \dots, F_{p}^{\mathcal{A}}$$ be the equivalence classes with respect to $$\sim$$ such that $$\# F_{1}^{\mathcal{A}} \geq \dots \geq \# F_{p}^{\mathcal{A}}$$. We call a triangle $$\triangle \in F^{\mathcal{A}}$$ facial if it is a face of $$\Gamma_{\mathcal{A}}$$, i.e., $$L \cap \text{int} \triangle = \emptyset$$ for all $$L \in \mathcal{A}$$. Denote by $$G^{\mathcal{A}} \subset F^{\mathcal{A}}$$ the set of all facial triangles, and we denote by $$G_{1}^{\mathcal{A}}, \dots ,G_{q}^{\mathcal{A}}$$ the equivalence classes with respect to $$\sim$$ such that $$\# G_{1}^{\mathcal{A}} \geq \dots \geq \# G_{q}^{\mathcal{A}}$$. Now we define $f(\ell) = \max_{\mathcal{A} \in \mathfrak{U}_{\ell}} \# F_{1}^{\mathcal{A}}, \quad \quad g(\ell) = \max_{\mathcal{A} \in \mathfrak{U}_{\ell}} \# G_{1}^{\mathcal{A}}.$ Additionally, we define $$F(\ell)$$ ($$G(\ell)$$, respectively) as the set of all integers $$u$$ such that there exists an arrangement on $$\ell$$ lines having exactly $$u$$ congruent triangles (congruent facial triangles, respectively). We write $$[s..t]$$ for the set of all integers $$u$$ such that $$s \leq u\leq t$$, put $$H$$ for $$F$$ or $$G$$, and $$h$$ is equal to $$f$$ if $$H=G$$ or $$g$$ if $$H = G$$. Whenever $$H(\ell) = [0..h(\ell)]$$, we say that $$H(\ell)$$ is complete.
Theorem 1. One has $$f(5) = g(5) = 5$$ while $$F(5)$$ and $$G(5)$$ are complete.
Thereom 2. One has $$f(6) = 8, \, 6 \leq g(6) \leq 7$$, $$F(6)$$ is complete, and $$[0..6] \subset G(6)$$.
Theorem 3. One has $$f(7) = 14, \, 9 \leq g(7) \leq 11, [0..10]\cup\{14\} \subset F(7)$$, and $$[0..9] \subset G(7)$$.
Theorem 4. One has $$16 \leq f(8) \leq 22, \, 12 \leq g(8) \leq 15$$, $$[0..16] \setminus \{13\} \subset F(8)$$ and $$[0..12] \subset G(8)$$.
In the last section, the author formulates some natural conjectures, for instance the author predicts that $$g(6)=6$$.

##### MSC:
 52C10 Erdős problems and related topics of discrete geometry 52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)
##### Keywords:
arrangement of lines; congruent triangles
OEIS
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##### References:
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