Extended Morley Theorem

I have occuppied myself with Morley's Theorem and researched its application in the realm of synthetic geometry. So extended, Morley's Theorem reveals several properties that might, on account of their geometrical beauty, be of interest to many actively pursuing this science.

Fig.1 

 

 

illustrates Morley's Theorem. All three angles of the triangle ABC are trisected resulting in the equilateral triangle LMN. Next we will extend the application of Morley's Theorem into projective space.

 

Fig.2 

 

shows the same triangle ABC, its lines extending on all sides and so creating the three 'outer triangles'. One of these is shaded in.

 

Fig.3

 

the angles of the same outer triangles are trisected. The result is again an equilateral triangle created through the intersection of the adjacent trisectors.

Fig.4 

 

applies Morley's Theorem to all three outer triangles. The result is three equilateral triangles. (An awesome sight, although expected).

Fig.5 

 

repeats the same construction as Fig.4, omitting all trisectors and so highlighting the aesthetic and structural aspect of the Theorem.

 

Fig.6 

 

What is surprising though is the fact that the triangles are interrelated, that two sides of each triangle are shared by the other two when extended, forming in their middle a fifth equilateral triangle integrating the outer three and encompassing the inner one. (Triangle of Integration).

Fig.7 

 

shows the same phenomena for an isosceles triangle (note the beautiful symmetry).

 

 

Fig.8 

 

shows the Theorem applied to a rectangular triangle. Note the same size of the triangle of integration with the equilateral triangle opposite the hypotenuse, and also the fact that adjacent trisectors above the hypotenuse meet in a right angle (i.e. on the circumference of ABC) while the two remaining trisectors are parallel. (Factors important for the construction shown in the Appendix).

 

 

Fig.9 

 

concludes the sequence with the equilateral triangle.

 

Extended Morley Theorem 03.jpg