On n-sectors of the Angles of an Arbitrary Triangle

Morley’s theorem shows that the three points, each of which is the intersection of the two internal trisectors that are the closest to the same side of an arbitrary triangle Δ , form an equilateral triangle. This beautiful theorem was proved mechanically by Wen-tsün Wu (J. Syst. Sci. Math. Sci. 4:207–235, 1984) in its most general form: the neighbouring trisectors of the three angles of Δ intersect to form 27 triangles in all, of which 18 are equilateral triangles, called Morley triangles. A natural question is: does there exist any equilateral triangle, other than Morley triangles, which is formed by three intersection points of the neighbouring angular n-sectors of Δ for n> 3 ? In this paper, we approach this question using specialized techniques with interactive, semi-automatic algebraic computations and prove that for n= 4 and 5 the three points, each of which is the intersection of the two internal (or two external) angular n-sectors closest to the same side of Δ , form an equilateral triangle if and only if Δ is equilateral. The computational approach we present can also be applied to other cases for specific n. How to establish the non-existence of equilateral triangles formed by the intersection points of angular n-sectors for general n is a question that remains for further investigation.