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On rational Morley triangles. (English) Zbl 0951.51009

The authors consider interesting subcases of the famous Morley theorem from elementary plane geometry. This theorem states that the trisectors of the angles of a triangle ABC, which are chosen in pairs neighbourly to one side of ABC in each case, meet at the vertices of an equilateral triangle, the “Morley triangle” of ABC. It is also well known that certain extensions of this figure yield even 27 “Morley triangles”, 9 of which (the so-called Guy Faux triangles) are generated by trisectors of only two angles of ABC, leaving 18 genuine, pairwise homothetic “Morley triangles”. Regarding these 18 triangles the authors prove the following theorem: If a rational edged triangle has a rational Morley triangle, then either the original triangle is equilateral (and 6 of the 18 Morley triangles are rational), or it is Pythagorean belonging to a one-parameter family (and 2 of the 18 Morley triangles are rational), or it belongs to a two-parameter family of triangles (and all 18 Morley triangles are rational).

MSC:

51M04 Elementary problems in Euclidean geometries
11D41 Higher degree equations; Fermat’s equation
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