This is Luzia's proof of the Pythagorean theorem via half-angle formulas: https://www.cut-the-knot.org/pythagoras/Proof109.shtml
I shared my essay "The Theoretical Importance of Half-Angle Formulas" on Reddit: https://www.reddit.com/r/math/comments/16ojbnd/the_theoretical_importance_of_halfangle_formulas/
#math #geometry #trigonometry #halfanglesmatter #symmetrymatters
Problem 1315 from Gogeometry 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
If the half-angle formulas aren't considered fundamental, then how did Viète manage to derive his formula for π using them? And why is it that the Weierstrass substitution, hailed as the most ingenious substitution in the world by Spivak, is actually based on those very formulas? If these half-angle formulas lack significance, then how on earth have I successfully derived the formulas of Heron, Brahmagupta, and Bretschneider, as well as the laws of cosines, sines, and tangents, the Mollweide formula (or perhaps we should credit Newton), the angle bisector formula, the inradius formula for mixtillinear incircle, and the sum and difference of angle identities, not to mention Euler's remarkable triangular inequality, all stemming from these supposedly non-fundamental formulas? And if that's not enough, how have I managed to extend the Pythagorean trigonometric identity (with over 2000 years of history), the Mollweide formula (or Newton's, if you prefer) (with over 300 years of existence), and even the half-angle formulas themselves (likely over 2000 years old) into broader generalizations? Still skeptical? Well, I invite you to click on the link below and see for yourself.
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 #ChatGPT
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 below 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
Yo defendiendo que las fórmulas de medio ángulo son más fructíferas que la ley de cosenos.
Emmanuel José García
