Moving Picture for Morley's general theorem
- Spiridon Kuruklis
Morley's general theorem
Morley's general theorem relies on his observation " in a triangle, the trisectors proximal to a side intersect on three sets of three parallel lines forming equilateral triangles" from where he inferred his famous theorem. It may be stated as: In any triangle, the trisectors of a type, proximal to sides respectively, meet at the vertices of a corresponding equilateral. The sides of the three equilaterals are parallel in pairs while the others meet at the intersections of trisectors proximal to a side of the triangle. More generally, in a triangle, the angle trisectors of same kind for all, a distinct kind for each, or a kind for one and its corresponding kind for the other two, proximal to sides respectively, meet at the vertices of an equilateral. This generalizes Morley's theorem by considering the interior and also the exterior and explementary trisectors. Thus the intersections of trisectors, proximal to sides respectively, form 27 equilaterals. 18 by trisectors of all three angles, called Morley triangles, and 9 by trisectors of two angles, called Guy Faux triangles. The Morley triangles formed exclusively by intersections of the interior, exterior, and explementary trisectors are called inner, central and peripheral triangle respectively.