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Some extremal problems in geometry. III. (English) Zbl 0328.05018
Proc. 6th southeast. Conf. Comb., Graph Theor., Comput.; Boca Raton 1975, 291-308 (1975).
[For the entire collection see Zbl 0313.00004.]
In Part I, and II [both authors, J. combinat. Theory Ser. A 10, 246-252 (1971; Zbl 0219.05006) and second author, Discrete Math. 7, 305-315 (1974; Zbl 0283.05008)] the authors discuss the maximum number of times $$f^a_k(n)$$ that the same non-zero area can occur among the triangles $$\Delta X_iX_jX_l$$, $$1 \leq i<j< \ell \leq n$$, where the maximum is again taken over all choices for $$X_1, \ldots ,X_n$$ in $$E_k$$. In this report they discuss the maximum number $$f^i_k(n)$$ of isosceles triangles that can occur (congruent or not), the maximum number $$f^e_k(n)$$ of equilateral triangles that can occur, the maximum number $$f^c_k(n)$$ of pairwise congruent triangles, and the maximum number $$f^s_k(n)$$ of pairwise similar triangles that can occur. All of these problems were posed at the end of Part I.

##### MSC:
 05B25 Combinatorial aspects of finite geometries 51M99 Real and complex geometry