Applications of Morley's Theorem

In summary, the conversation discusses the exploration of Morley's Trisector Theorem and the struggle to find real life applications for it. The speaker notes that while the theorem is interesting, it appears to have limited practical uses. They ask for suggestions on how to relate the theorem to the real world, and mention that even seemingly abstract theorems have been found to have practical applications. The conversation also includes links to further reading on the topic, and mentions that other geometric theorems such as Pythagoras, Thales, and sine and cosine theorems are more commonly used in real world applications.
  • #1
Cheesycheese213
55
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I was planning to do an essay exploring Morley's Trisector Theorem, but I was having trouble finding and real life applications of it?

It states that "in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle".

I thought it was pretty interesting so I chose it, but after researching it more, it looks like there really isn't much use for it other than being one of those "beautiful results" that are just interesting but can't really be used? One of the criteria is to reflect on the essay and what we learned, and we were told that we could for example relate it to the real world. But, if I can't even apply it to math I'm not very sure how I could do that?

Is there something (real life or in math) that the theorem can be applied to? Thanks!
 
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  • #2
This is as with many geometric theorems. They appear as rather abstract and useless, but surprisingly appear from time to time. If you asked me for an application of Thales, I would have to admit, that I don't know one. Nevertheless I used it very often and I'm sure, e.g. construction engineers use it, too. One thing that came to mind were snow chains. I don't know which underlying principles they use for manufacturing them, but I could imagine that Morley could be used in some force diagrams.

You could have chosen worse:
https://www.emis.de/journals/BAG/vol.43/no.2/b43h2sta.pdf
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.38.4801&rep=rep1&type=pdf
but Pythagoras, Thales, sine and cosine theorems are certainly the frontrunners of all geometric theorems with respect to their applications.
 
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1. What is Morley's Theorem?

Morley's Theorem, named after mathematician Frank Morley, states that in any triangle, the three points of intersection of the adjacent angle trisectors of each angle form an equilateral triangle.

2. What are the applications of Morley's Theorem?

Morley's Theorem has various applications in geometry, trigonometry, and algebra. It can be used to solve problems related to angle trisectors, construct regular polygons, and even in the study of elliptic curves.

3. How is Morley's Theorem useful in real life?

In real life, Morley's Theorem can be used in architecture and design to create symmetrical and aesthetically pleasing structures. It can also be applied in navigation and surveying to determine angles and distances.

4. Are there any limitations to Morley's Theorem?

Yes, Morley's Theorem only holds for triangles in Euclidean geometry. It does not apply to triangles in non-Euclidean geometries such as hyperbolic or spherical geometry.

5. How can Morley's Theorem be proved?

Morley's Theorem can be proved using various methods, such as Euclidean geometry, trigonometry, and complex numbers. One popular proof involves using the Law of Cosines and the properties of equilateral triangles.

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