Triangles In Arrangements Of Lines

An arrangement of lines is the partition of the plane formed by a collection of lines.An arrangement of lines A shall be a finite family of l lines L1,…,Ll.In this paper,we mainly study the maximum number of congruent triangles and similar triangles in finite arrangements of l lines in the Euclidean plane.In Chapter 1 we introduce some basic concepts related to the content of this thesis.In Chapter 2 we first prove C.T.Zamfirescu's conjecture in[23]:the maximum number of congruent facial triangles in finite arrangements of 6 lines in the Euclidean plane is 6,g(6)?6 and G(6)is complete.Then we prove that the g-optimal arrangements of 6 lines are neither c-unique nor g-unique.Finally,we prove that the maximum number of congruent facial triangles in finite arrangements of 7 lines in the Euclidean plane is at most 10,and have 9 ? g(7)? 10.In Chapter 3 we generalize the problem of congruent triangles in arrangements of lines to study the maximum number of similar triangles in arrangements of lines.For the case of 3 ? l ? 6,we determine the maximum numbers of similar triangles and similar facial triangles in finite arrangements of l lines in the Euclidean plane.For the case when l is even and not less than 8,we show that the lower bound of the maximum number of similar triangles in finite arrangements of l lines is 3l.