Trisectors like Bisectors with equilaterals instead of Points

Abstract

It is established that among all Morley triangles of â–³ABC the only equilaterals are the ones determined by the intersections of the proximal to each side of â–³ABC trisectors of either interior, or exterior, or one interior and two exterior angles. It is showed that these are in fact equilaterals, with uniform proofs. It is then observed that the intersections of the interior trisectors with the sides of the interior Morley equilateral form three equilaterals. These along with Pasch‘s axiom are utilized in showing that Morley‘s theorem does not hold if the trisectors of one exterior and two interior angles are used in its statement.