Dear Dr. Teia,

I have decided to accept your paper for inclusion in The Journal of Open Engineering following your revisions in response to the reviewer comments.

Regards,

Devin

GENERAL COMMENTS

It is clear that the author has a keen sense and love for geometrical proof, which is welcome. This article provokes thought about when an algebraic or analytical result can be proved by geometric means.

However, from an Engineering point of view, I regret to say that I do not see a lot of focus here. The examples from Civil and Electrical Engineering are at best of tangential interest, as the situations presented do not have any special affinity to the 120-degree angle, and so these examples provide no motivation for why this particular special case is important. There might be some examples in engineering with high degrees of symmetry that are more relevant, but even then, I cannot think of any situations in which knowing the direct result of the Law of Cosines for the 120-degree angle is anything more than a nice convenience in the presence of much more complex issues.

From an Engineering Education point of view, there is possible relevance - maybe it is useful to teach students to apply the 'short cut' of the Law of Cosine for the 120-degree case. But from my perspective having taught Statics for over 20 years, the students who struggle - and there are many - have deficiencies in algebra, geometry, and trigonometry. The proofs in this paper would escape the grasp of even most well-performing Statics students, and I think they would miss the central idea that ‘pure’ geometry can sometimes be used to derive analytical results. Moreover, emphasizing the single case for 120 degrees would distract from the more basic necessity, which is to help students learn how to confidently apply the Laws of Sine and Cosine in general cases.

From a historical perspective, there is some interesting Mathematics history to which the article alludes (see comments below), but I cannot think of anything analogous from the history of Engineering.

TECHNICAL SOUNDNESS

For the reasons stated below, I did not attempt to verify most of the equations. However, there is at least one important error, which occurs in equation (2): the (-) should be (+).

CLARITY

I found the proofs to be tortured and lengthy to the extent that I did not attempt to trace through most of the details. Instead, motivated by the author’s approach, I discovered a much-simplified proof based on adding a few line segments to the Figures 10(a) and 10(b); in Figure 10(b), the “*z*-triangle” can be inscribed in a larger equilateral triangle of sides *x *+ 2*y*. I would be happy to discuss this with the author upon request.

There are numerous grammatical, typographical, and formatting errors, which I will not comment on explicitly, as I think there are more fundamental concerns with the article before reaching the point of editorial refinement. But one general comment is that all mathematical characters should be set in a standardized typeset, and all equations should be numbered.

COMPLETENESS

The paper is complete in the sense that it presents an approach for deriving a result and completes a proof.

However, very problematic is at least one instance of plagiarism. The following segment is a direct phrase match from the reference [3]:

“Thus, Euclid's Elements (300 B.C.) contains the seed of concepts that lead to the law of cosines. In the fifteenth century, the Persian astronomer and mathematician al-Kashi provided accurate trigonometric tables and expressed the theorem in a form suitable for modern usage.”

Although [3] is cited immediately following as a reference, there is no acknowledgment of the quotation. This MUST be corrected. I will also comment that this brief historical account, which jumps from Euclid to al-Kashi, is similar to the account in Wikipedia, whereas in [3], there is somewhat more history given. I have not studied this question in detail, but I would imagine that there is further history on the geometrical basis of the Law of Cosine that could be reported.

It is also both strange and inappropriate that the author did not cite another of his own, published works on an essentially similar topic, “Extended Pythagoras Theorem Using Hexagons” (https://ccsenet.org/journal/index.php/jmr/article/view/0/46329). I discovered this article because some near-phrase matches were detected from this article when I ran the manuscript through Ithenticate. Although the article on “ … Using Hexagons” uses different details to deduce the Law of Cosine for the different case of a 60-degree angle, vs. the 120-degree angle in the manuscript, the overall scheme is very similar, and requires citation.

OPENNESS AND REPRODUCIBILITY

This article appears to meet the openness and reproducibility standards. The references that I attempted to check were openly accessible (although I don’t think that cited works should be required to be open-access, despite the mission of this Journal).

GENERAL COMMENTS

I thank the reviewer for taking the time and effort to present his opinion on the content of this article, and for all the valuable comments.

I am afraid I do not agree with the reviewer on the first point, as there are several cornerstone examples of interest available. Three-phase electric power is popularly employed in electrical engineering, and operates by setting currents of combined common frequency systems out of phase by 120 degrees, and I quote (link directly below) “In electrical engineering, a three-phase system indicates a combined system of 3 alternating current circuits (for a system of production, distribution and consumption of electricity) that have the same frequency.” A sentence was added at the beginning of section 4.2 to acknowledge this, along with an appropriate reference.

https://solar-energy.technology/electricity/electric-current/three-phase-system

A second exemplary reference: https://electronics.stackexchange.com/questions/470501/help-on-another-triphasic-exercise

A third and fourth reference: https://www.geistglobal.com/sites/all/files/site/geist_ep901_-_three-phase_electric_power_distribution_v2.pdf

https://www.sciencedirect.com/science/article/pii/B9780128044483000104

Another example of an important application in electrical engineering is telecommunications, in particular cellular base station antennas deployed in an array along three faces arranged in a triangular manner ― with each face having typically 3 or 4 radiating elements (see link below on page 336; and the second link for a photo of a real triangular antenna station). The analysis done to the AC generator/motor is valid by replacing the rotating magnet with a transmitting mobile phone, and the coils by each face of the triangular antenna array. The directivity, and hence intensity of signal strength perceived by two of its three antenna faces disposed at 120 degrees to each other will vary as the phone rotates around the antenna (by the displacement of the individual, as it transits), recording the same reduction/increase in signal behavior as that explained before for the AC generator/motor application (where the magnet rotated within two coils angular-spaced by 120 degrees). This has been added to section 4.2.

https://news.launch3telecom.com/what-is-an-antenna-array/

Favoring complex issues over simplicity is ― in this author’s opinion ― the first step to overlooking the great discoveries awaiting when adopting a simple approach to investigate any given challenge. Here the book “Small is beautiful” conveys the right philosophical message herein intended.

https://en.wikipedia.org/wiki/Small_Is_Beautiful

Moreover, the simple fact that the present article determines that the variant equation interconnects triangles, is in itself something that was not known (let alone proven) by simply applying the law of cosines, in its present form and understanding. The present understanding of the law of cosines is incomplete, as the geometrical linkage between the areas in the equation is not known or proven for any angle gamma, except for 90 degrees in the Pythagoras theorem connecting squares (with its own various geometrical and mathematical proofs), and now for 60 and 120 degrees in the new variants of this theorem connecting triangles and hexagons. I leave open the challenge of extending these proofs for gamma equal to 108 degrees, which interconnects the area of pentagons.

Throughout science, it is known that studying fundamental systems is the precursor to perceiving applications to more complex end situations. One cannot do an application without first formulating the theory (the basis for any PhD thesis). The present case is no exception, and the first step is completely fulfilled by presenting the novel theory using triangles. Then, it provides a potential application to improve the understanding of fundamental systems such as AC motor/generator and triangular antenna arrays, where no comments were provided invalidating the present logic and conclusions. Other more complex end applications are out of the scope of this article.

Another simple example in history where theory precedes application is the Taylor series that was invented in the 18^{th} century by Brook Taylor, and found centuries later many of its most critical applications in physics ― such as Numerical Analysis & Methods (for example the Navier-Stokes equations used in Computational Fluid Dynamics), Quantum & Nano Physics and Calculators.

https://waterloostandard.com/post/taylor-series/

Not to mention Einstein’s Theory of Relativity that was opposed by many preeminent scholars by many years as being wrong (let alone it being useful with an application), and now it is the cornerstone of modern physics. Hence, in the authors perspective, not finding an application simply is a synonym of not trying hard enough, while (in this case) at the same time, of also not knowing enough to do so.

Another application of the present theory that is foreseen by this author in the future ― but it is too extensive and merits an article of its own ― is in electronics. The RLC circuit operates based on the principle of the Pythagoras theorem (via the same dynamics that govern a mass-spring-damper system), hence it is possible to conceive a variant circuit that operates based on the new theorem, something that this author has named *coupled-RLC circuit*. Again, this is an application that the reviewer cannot know about in the open literature of electrical engineering, simply because this author has not yet published it.

This article does not intend to teach students of existing trigonometric knowledge (that is not its purpose), as this is well covered by existing literature. It is the intention of this article to push the boundaries of the known trigonometry into the unknown, and that may be discomforting and disorientating to anyone uninitiated to such a new topic ― whether it be a student or a teacher. The fact that teachers spend decades teaching the same thing, sure compels and welcomes a distraction with a novel fresh idea into the possible future of trigonometry. Naturally, nothing new comes from doing the same thing repeatedly. To this respect, the present work is highly necessary to both mathematical research and teaching, and its natural subsequent application to engineering.

CLARITY

A common problem found in mathematical proofs is the lack in clarity of the steps taken (a struggle that mathematical genius Ramanujan highlighted with his own life and work experience). Hence at the expense of extra attention to detail, this proof is published with high precision. This obviously requires more length in explanation, and consequentially more attention from the reader, which is a good thing as it compels her/him to expand its own capabilities to interpret the inherent complexities of mathematics. Further simplified versions of this proof are possible for popularized explanations ― like for teaching ― in the future to come, however the initial proof needs to be as precise as possible for completeness in capturing the totality of the proof.

If the reviewer has found a variant of this proof, the best venue would be for him to submit it to a peer-reviewed journal publication, of which I reciprocally would be happy to be a reviewer, and assess the truth and usefulness of its content.

Referring back to the need for precision and proofs, if these mentioned grammatical, typographical and formatting errors are not highlighted by the reviewer (due to a matter of convenience), then there is no proof that they exist. This is strange, as the article was initially written in MS Word which has a spell checker, passed to TJOE which also has its own spell checker with a semi-automatic HTML text formatting, and the other reviewer (who demonstrated to be quite thorough) only highlighted a few typos that are already corrected. Given the low input, the most that can be done is to pass the article again through a spell checker (no errors were mentioned from the actual mathematical and geometrical content), and see if something has been overlooked. Upon conclusion of this task, nothing new was detected.

COMPLETENESS

The sentence in question ― the result of a simple overlook ― was paraphrased and source cited to meet the referencing standards of MIT online publication on academic integrity (link below) .

https://integrity.mit.edu/handbook/what-plagiarism

As for the Wikipedia comment, by doing the appropriate investigation, one can find that the source of the quoted statement ― which is presented both in this article and in the Wikipedia page ― is referenced to the book from Clifford A. Pickover, which is properly identified in both this article (as reference [3]) and in Wikipedia. The amount of history provided on the law of cosines is deemed sufficient for the purpose of this article. Hence, introducing a further statement that more history exists on the law of cosines, without making a proper investigation and/or referencing, can only be regarded as inconclusive speculation.

https://en.wikipedia.org/wiki/Law_of_cosines

The reason why my most recent work “Extended Pythagoras Theorem Using Hexagons” was not yet referred is simple. Noticing back to its release date (November 21, 2021), it is seen that it is later than that of the release of the present draft (November 15, 2021), hence the first not appearing in the second. Being of equal importance and relevance to the present work, the hexagonal proof is now remarked (as originally intended by this author) as an added sentence at the end of section 1.2, and appropriately referenced in the final version of this publication as reference [4].

Thank you for acknowledging that the present article meets the standards of this journal.

I would like to thank once again the reviewer for his time and effort in contributing to the assessment of this article, and I wish him a prosperous future.

Best regards,

Luis Teia, PhD

Dear Tim,

I have completed answering your recommendations to the best of my abilities (the answers were posted in the original article, while the rectifications were issued as a new draft - see links below). Please have a look to see if there is any point you wish me to further elaborate. I have tried to be methodical.

Answers in original article -->> https://www.tjoe.org/pub/ei8ogu9d/release/1

New draft with corrections -->> https://www.tjoe.org/pub/ei8ogu9d/draft

It was a pleasure working with you.

Kind regards,

Luis Teia

I don’t think it’s *incorrect*, just not what you’ve gone with for your proof - which is focussed on equilateral triangles rather than squares (or any other shape).

To avoid confusion, the sentence was removed (in the new draft version). Cheers.

Proof has been completed when you get to (11). Agree that (12) is more convenient, but not the theorem you have set up.

Agreed. I have placed (in the new draft of the article) both the phrase “Conveniently…” and Eq.(12) below the horizontal line, resulting in the completion of the proof just below Eq.(11). Thanks

Should this be prior to the previous paragraph? Seems out of order to be using this in a formula before establishing the area of these 120-degree triangles.

Yes, I see your point and agree. The explanation of the area of the 120-degree triangles was moved up (in the new draft version) to just before “Concluding…”, just after the sentence “The 120-degree-angled triangle in Figure 9a EB’F becomes AB’B in Figure 9b.“ This way it is in the flow of the explanation, and before using it in any of the formulas. Cheers!

of

Term added (in the new draft)

Is Figure 8 intentionally drawn so that triangles KML and FLB appear approximately equal?

Not really, this was a coincidence. To avoid confusion, Figure 8 was slightly altered (i.e., triangle ABC was slightly rotated, with the remainder triangles and clockwork following this rotation) [in the new draft of this article] so that the difference between KML and FLB is more noticeable (i.e., generally they are different). This rotation is minor, and does not affect the validity of any of the explanations and arguments given.

I understand hypotenuse to only have meaning in the context of a right-angled triangle. Perhaps “longest side” would babe more appropriate?

The recommended correction was made (in the new draft) for all instances related to the triangle version of the theorem. Cheers.

Given the non-standard axes in use here, would it be worth stating that you’re setting up three axes, x, y and w at 60 degrees to one another? This will make establishing that triangles are equilateral more watertight.

Indeed, an early warning would facilitate the reader’s transition between theorems. I have added a paragraph (in the new draft article) just after the title of Chapter 3, for this specific purpose. Thanks.

How do we know this is equilateral? (I’m convinced that they are, but only intuitively.)

I see the concern. To resolve this, an explanation was added to the text (just before this sentence, and in the new draft version of this article), describing step-by-step the arguments that, emerging directly from the extension of segments parallel to each axis from the corners of the outer equilateral triangle DAB, these define amongst themselves the triangle KML as being equilateral. Cheers

shape also

Term added (in the new draft)

CLARITY

The tense doesn’t match the surrounding writing - “changes” instead

Tense changed as suggested (in the new draft)

CLARITY

It seems strange in Figure 5 to have the orthogonal axes positioned diagonally rather than vertical/horizontal, and it isn’t immediately obvious why it has been done this way (it makes sense after seeing Fig 7, but I’m still not convinced that it is the best way to represent this)

It’s a good point ! To make it easier for all to understand, the orthogonal axes in Figure 5 and 6 were altered back (in the new draft version of this article) to the traditional vertical and horizontal layout. An explanation is later introduced determining how these axes alter (in number and place), when we move to the triangular version of the theorem.

CLARITY

This doesn’t sound quite right - may be a translation issue?

This point was rewritten and further elaborated (in the new version of the article), for added clarity.

CLARITY

In Figure 6, below, it might be worth adjusting the rotation so that it is clear that point N’ is *not* on the circle.

Figure 6 has been changed to make it clear that point N’ is (in general) not on the circle. Cheers

CLARITY

Previously sides *x *and *y *were defined as the distances from A and B to O, which happened to be coincident with the orthogonal axes. It may be better to define these lines initially as being parallel to the orthogonal axes, and meeting at a point M, and then because in Figure 5 these lines are coincident with the orthogonal axes, then M=O.

Yes, I see your point and agree. I have altered (in the new draft of this article) the nomenclature such that the sides x and y for both theorems are related to distances from A and B to M — both parallel to the axes — for which the particular case x equal y coincides point M with the origin of the axes given by Point O — i.e., M=O — and for the general case x different from y, point M is not the same as point O. Thanks