It follows from GM / DG = cos(0.5 * γ) / cos(0.5 * α) and (MN / EN) = cos(0.5 * α) / cos(0.5 * γ), where α is the angle at A, and γ is the angle at C. Since DM = ME, the conclusion is easily deduced. You can find the detailed steps here: https://geometriadominicana.blogspot.com/2023/08/solution-to-problem-1315-of-gogeometry.html

#math #halfanglesmatter #geometry #trigonometry

If the half-angle formulas are not fundamental, how did Viète derive his formula for π based on them? How is it that the most ingenious substitution in the world (quoting Spivak), namely the Weierstrass substitution, is based on them? If the half-angle formulas are not fundamental, how have I been able to derive the formulas of Heron, Brahmagupta, and Bretschneider, the law of cosines, the law of sines, the law of tangents, the Mollweide formula (or should I say Newton's), the formula for the angle bisector, the formula for the inradius of a mixed-inscribed circle, the sum and difference of angle identities, Euler's triangular inequality, all from them? If the half-angle formulas are not fundamental, how have I been able to generalize the Pythagorean trigonometric identity (with over 2000 years), the Mollweide formula (or should I say Newton's) (with over 300 years), the law of tangents (with over 900 years), and even the very half-angle formulas themselves (with over 2000 years, probably), from them? Don't believe me? Check the link below.

🔗 [https://drive.google.com/file/d/1EvXiZT62a2oVWFWtW5NoAyQfUAi-JnFe/view?fbclid=IwAR30XmSx2-i9WTa27TtG6hEZe0Zui77JnUHGXEQ15aEO4Dmkh41V44SsLV4]

#math #fundamental #halfanglesmatter #symmetrymatters

I am grateful that the Ministry of Youth of the Dominican Republic has recognized my half-angled approach! You can see the details of my approach in the link below.

Perhaps young aspiring mathematicians are the best target for my work. The old mathematicians could be very prejudiced.

🔗 [https://drive.google.com/file/d/1EvXiZT62a2oVWFWtW5NoAyQfUAi-JnFe/view?usp=sharing]

#math #halfanglesmatter #symmetrymatters

Dear friends,

I hope this message finds you all in high spirits and ready for a thrilling mathematical adventure! Today, I am thrilled to share with you a groundbreaking essay that delves into the untapped potential of half-angle formulas and their astonishing applications.

🔗 [https://drive.google.com/file/d/1wihl_DAE547OZPji5ounyHRYm6d8SZ6Y/view?usp=sharing]

In this thought-provoking piece, titled "The Theoretical Importance of Half-Angle Formulas," I explore the remarkable world of these often-overlooked formulas and their far-reaching implications. But before you dive in, let me share a captivating quote by the legendary George Pólya: "An idea which can be used only once is a trick. If one can use it more than once, it becomes a method."

Now, why is this quote relevant to my work, you might ask? Well, my friends, in the realm of mathematics, finding a single result is indeed fascinating, but uncovering a method—a systematic approach that can be employed repeatedly—is truly extraordinary. My essay takes you on a journey through the complexities of discovering and harnessing methods using the half-angle formulas.

Warm regards,
Emmanuel José García.

#math #trigonometry #newapproach #polya #halfanglesmatter #symmetrymatters #essay

Method vs trick according to George Polya

ChatGPT agrees that my half angle approach is a method rather than a trick.

#newmethod #halfangleapproach #math #geometry #trigonometry #georgepolya #halfanglesmatter #symmetrymatters

As an amateur mathematician, I believe my most significant finding is the recognition of the central role played by half-angle formulas. Their significance goes beyond the domains of geometry and trigonometry in the plane. Despite their origins dating back to Ptolemy, no one, without exception, had noticed their fundamental connection to other crucial theorems in the systematic manner that I have. I am confident that Conway would have been delighted by my work.

#math #geometry #trigonometry #halfanglesmatter #symmetrymatters

The half angle formulas should be an object of study by the mathematical community. I don't know why some mathematicians keep quiet. Isn't the evidence I have presented strong enough? Or are you still in the prejudice that nothing significant can be drawn from classical geometry anymore?

#math #geometry #halfanglesmatter #symmetrymatters

Earlier I had said that the half-angle formulas are an essential ingredient in the derivation of Viète's \(\pi\) formula. According to Eli Maor, Viète's formula marks the beginning of mathematical analysis, and Jonathan Borwein calls its appearance "the dawn of modern mathematics".

The work described in the link is another example of the centrality and versatility of the half-angle formulas (although the authors mistakenly call them double-angle formulas), a centrality that transcends the field of geometry and trigonometry in the plane.

https://www.sciencedirect.com/science/article/pii/S0021904513001159?via%3Dihub#br000045

#math #geometry #trigonometry #halfanglesmatter #symmetrymatters

My wake-up call for the extraordinary explanatory power of half-angle formulas has reached 3,000 views on MathSE. Are the half angle formulas more fundamental than the Pythagorean theorem or the law of cosines?
#math #trigonometry #geometry #halfanglesmatter #symmetrymatters

Yo defendiendo que las fórmulas de medio ángulo son más fructíferas que la ley de cosenos.

#math #trigonometry #halfanglesmatter

But what excites me the most about these formulas is their potential to revolutionize the way we teach trigonometry to high school students. By introducing the half-angle formulas in their advanced form, we can help students better understand the development of the formulas that derive from them. This is where the half-angle formulas truly shine and can make a significant impact on math education.

GeoDom: The Half-Angle Formulas: A Powerful Tool for Trigonometry Students http://geometriadominicana.blogspot.com/2023/03/the-half-angle-formulas-powerful-tool.html?m=1

#math #trigonometry #halfanglesmatter #education #newapproach

Viète's formula for \(\pi\), the Weierstrass substitution and my approach based on half-angle formulas are three examples of how useful it can be, when you think about mathematics, to obey certain symmetries.

#math #symmetry #halfanglesmatter

Had I lived in the 18th century, it is likely that my approach to metric relationships in Euclidean geometry, based on half-angle formulas, would have been ubiquitous in high school textbooks, articles, and reviews on the subject nowadays.

#math #halfanglesmatter #symmetry #geometry

Initially I had found an extension of Lami's Theorem for cyclic quadrilaterals, but on the PhysicsSE forum, Philips Wood was dissatisfied and challenged me to generalize Lami's Theorem to a general quadrilateral. So I did it.

Link: https://math.stackexchange.com/questions/4490209/generalizing-lamis-theorem

#math #phisics #halfanglesmatter

Like Josefsson, I'm almost sure Euler would have loved my way of approaching metric relationships in Euclidean geometry using the half-angle formulas.

#math #geometry #halfanglesmatter

Geometric configurations that obey a certain symmetry are the ones that almost always (read always) produce interesting results. Personally, I have never seen an interesting result arise from an asymmetric geometric configuration. Can someone give me an example in classical geometry?

#math #geometry #halfanglesmatter

They are.

#math #halfanglesmatter

My generalization of Newton's formula has been accepted for publication 😳. The generalized formula in the image extends Newton's formula to cyclic quadrilaterals. The link below contains Newton's original formulas.

Link: https://mathworld.wolfram.com/NewtonsFormulas.html

#math #geometry #halfanglesmatter

Trigonometry has given us much more than number theory. Let's make trig great again!

#math #halfanglesmatter

@johncarlosbaez In the image you can see why. BTW, the standard half-angle formulas can be obtained from the ones in the image by substituting from the law of cosines (which is also another way of writting the half-angle formulas). So they are different forms of the same thing. If you have time, also take a look of this: https://hsm.stackexchange.com/questions/15010/the-role-of-symmetry-in-mathematics-and-the-half-angle-formulas

#math #halfanglesmatter #symmetry