Abstract
We study the maximum number of congruent triangles in finite arrangements of l lines in the Euclidean plane. Denote this number by f (l) . We show that f (5) = 5 and that the construction realizing this maximum is unique, f (6) = 8 , and f (7) = 14 . We also discuss for which integers c there exist arrangements on l lines with exactly c congruent triangles. In parallel, we treat the case when the triangles are faces of the plane graph associated to the arrangement (i.e. the interior of the triangle has empty intersection with every line in the arrangement). Lastly, we formulate four conjectures.
Highlights
A problem from mathematical folklore asks for bounding four congruent triangles with six matchsticks
Our main focus lies on constructing planar arrangements in which a fixed number of lines bound as many congruent triangles as possible
We can identify LA1 with LB1 and LA2 with LB2 such that an arrangement C is obtained in which, seeing A and B as sub-arrangements of C, no good triangle lies in both A and B . (Note that the number of good triangles in C may be larger than the sum of the number of good triangles in A and B, see e.g. Figure 8, in which the arrangements from Figures 1 and 5 (b) are joined: the original arrangements have 14 and 8 good triangles, respectively, but the new arrangement has 26.)
Summary
A problem from mathematical folklore asks for bounding four congruent triangles with six matchsticks. A triangle in A ∈ A shall be the convex hull of the set of intersection points of three non-concurrent pairwise non-parallel lines in A. (Note that, as mentioned above, we do not consider line arrangements in which all lines meet in a single point.) A ∈ A is c-unique if there exists no B ∈ A such that A and B are (a) not combinatorially equivalent and (b) |H1A| = |H1B|, where H is F or G. F ( ) (G( )) is defined as the set of all integers u such that there exists an arrangement on lines having exactly u congruent triangles (congruent facial triangles).
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