so I was reading this

- Two US high schoolers believe they have cracked a mathematical mystery left unproven for centuries.
- Calcea Johnson and Ne’Kiya Jackson looked at the Pythagorean theorem, foundational to trigonometry.
- The American Mathematical Society said the teenagers should submit their findings to a journal.

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**chat gpt **

*Two high school seniors from New Orleans claim to have proved the Pythagorean theorem using the Law of Sines, which does not rely on circular trigonometry. The theorem has been around since ancient Greek times and defines the relationship between the three sides of a right-angled triangle. Although the theory holds true in every plausible example, no mathematician has been able to establish its truth from first principles. The American Mathematical Society has encouraged the young mathematicians to submit their findings to a journal for assessment.*

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# Pythagorean theorem

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From Wikipedia, the free encyclopedia

Type | Theorem |
---|---|

Field | Euclidean geometry |

Statement | The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). |

Symbolic statement | �2+�2=�2 |

Generalizations | Law of cosinesSolid geometryNon-Euclidean geometryDifferential geometry |

Consequences | Pythagorean tripleReciprocal Pythagorean theoremComplex numberEuclidean distancePythagorean trigonometric identity |

In mathematics, the **Pythagorean theorem** or **Pythagoras’ theorem** is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides *a*, *b* and the hypotenuse *c*, often called the **Pythagorean equation**:^{[1]}�2+�2=�2,

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https://meetings.ams.org/math/spring2023se/meetingapp.cgi/Paper/23621

##### Abstract

In the 2000 years since trigonometry was discovered it’s always been assumed that any alleged proof of Pythagoras’s Theorem based on trigonometry must be circular. In fact, in the book containing the largest known collection of proofs (The Pythagorean Proposition by Elisha Loomis) the author flatly states that “There are no trigonometric proofs, because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean Theorem.” But that isn’t quite true: in our lecture we present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry—the Law of Sines—and we show that the proof is independent of the Pythagorean trig identity \sin^2x + \cos^2x = 1.

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In trigonometry, the **law of sines**, **sine law**, **sine formula**, or **sine rule** is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,�sin�=�sin�=�sin�=2�,

where *a*, *b*, and *c* are the lengths of the sides of a triangle, and *α*, *β*, and *γ* are the opposite angles (see figure 2), while *R* is the radius of the triangle’s circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;sin��=sin��=sin��.

The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the *ambiguous case*) and the technique gives two possible values for the enclosed angle.

The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines.

The law of sines can be generalized to higher dimensions on surfaces with constant curvature.^{[1]}

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Law of sines was mentioned so I added it her

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and the law of sines

I hope these two get scholarships and at least a good future..

Enjoy

Mathing..

lol