Morley's Theorem, a Proof

Well, the diagram may be quite useful.

Let's check the first paragraph:

"All triangle endpoints are formed by the intersection of three distinct lines. That is, for three triangle sides A, B, and C, the endpoints are generated as follows: ∪{(A∩B) , (B∩C) , (C∩A)}. The endpoints are generated not only by the intersection of each pair of lines, but equivalently by the intersection of the lengths of each of their reversed angular projections, thus each may be equally replaced by their equivalent angles in the set, which is stated as ∪{∩[α, β, γ]}."

1. I believe understand ∪{(A∩B) , (B∩C) , (C∩A)}, viz., three lines intersect in three points and a triangle is thought of as the union of three points. This is a strange concept that omits the sides of the triangle, but I can provisionally accept it.

2. I believe I understand the concept of the "reverse perspective." I do not understand how does it swap the roles of angles and lines. A diagram may help here.

3. I do not understand the distinction between lines and lengths. If there is any it should be explained. If there is none, the terminology should be more reserved.

4. I do not understand ∪{∩[α, β, γ]}.

   α, β, γ are the angles of a triangle understood geometrically. ∪{∩[α, β, γ]} is the union of three plane regions. If that is right whose intersection is taken? What is left to intersect?

With regard to the first case, the points are considered exclusively, as a given triangle's sides may be subtracted while retaining its endpoints without losing its shape, as line segments connecting the points are secondary information which depend only the existence of the two mutual endpoints which connect them. This lack of dependence on limited line segments defining the triangle allows more freedom to illustrate only the key connections.

Second, rather than considering an angle as a secondary product dependent on the intersection of two lines, the reverse perspective of having a given angle first, and then resultantly generating the two lines it projects changes a given angle from a dependent variable into an independent variable, which is key to showing how a given set of angular relations can by itself produce another given set of line intersections.

Third, a length refers to the lines of intersection only in the sense in which they are the dependent variable generated by a given angle's projection, and they are different in this sense from the line segments which bound the triangle in that they are considered to extend infinitely and are only limited by their mutual intersection with each other, as in the case of a triangle. For example, a triangle with vertex A has angle a, and vertex B has angle b, and if it is viewed as being relatively upright with A at the top and B to the bottom right, the left length of B intersects with the left length of A, which produces the third vertex C; B's left length would have gone on infinitely without the intersection with the left length of A; a triangle is defined by the intersection of its angle projections in this way, and the lengths are meant to extend indefinitely until intersection with another length for the later purposes of the proof, with the term length reserved to specifically refer only to the line of the angular projection for clarity purposes.

Fourth, as mentioned above, it is the intersection of the angular projections which defines the endpoints of the triangle. A's angle "a" projection extends on the right side to coincide with vertex B's angle "b" projection on the right side, and vertex C's angle "c" projects to the right side to coincide with the projection of "a" and the origin point of the projection of angle "b" on the right side; it is at that point that all three angle projections' lengths coincide and intersect, and the same is for the other two vertices of the triangle, and the intersection of all three projections then occurs only at those three points, which are the three endpoints of the triangle.

The rest of the proof follows incontestably from the proper establishment of those principles, which themselves simply amount to a new, more elastic perspective of equally defining otherwise traditional concepts. The main insight was the realization that the sum of one of each of the trisected angles of any given triangle always add up to 60 - (3a + 3b + 3c)/3 = a + b + c = 180/3 = 60 - 60 being the measure of each angle in an equilateral triangle as well; thus these trisected angles together can represent each of the equilateral triangle's angles, which becomes 3(a + b + c) = 3a + 3b + 3c, which are the original angles. The key was simply to show the proper transformations which brought this about, part of which involved transforming the angle of a triangle from a dependent variable to an independent variable. Please let me know if you think the information presented here makes the proof more clear.

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Second, I can live with a notion of angle as a combination of two intersecting lines or rays, if you prefer. Is that any different from your idea of reversed angular projection? Somehow, the word "projection" has in my mind an association that requires to indicate what is projected and to where. There is no gain in using that word. So you like to see a triangle as an intersection of angles, do you? Or just the lines forming those angles? What does ∪{∩[α, β, γ]} relate to? This union/intersection symbolism appears to play an important role in your way of thinking. Just explain please this a step at a time:

What is ∩[α, β, γ]?
What is ∪{∩[α, β, γ]}, as related to the above?

Third, if you do not mind, I believe one can talk without confusion of "sides of a triangle" and "side lines of a triangle". A side is a line segment; a side line is a line (an object extending indefinitely in two directions). You use the word "length" in the sense of "expanse" which you could but which is not strictly necessary. "Length" has commonly a different association. It's rather a number or a measure. This is quite unlike the word "circumference" that is used both to denote the length of a circle's border and the border itself.

Once again, thinking of an angle as projecting its side lines is unnatural to my world view. Why? Let it just have, possess, consist of, comprise, or whatever else, its side lines. Angle may well be an abstract, Platonic object with incarnations in a common plane - other geometric objects constitute distinct abstract concepts and have their plebeian realizations that we approximate by drawing in the plane. It is unusual (and, to me, confusing) to refer to such an association of abstract and concrete a projection.