Articles containing proofs | Theorems about quadrilaterals and circles | Euclidean plane geometry
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. If the vertices of the cyclic quadrilateral are A, B, C, and D in order, then the theorem states that: where the vertical lines denote the lengths of the line segments between the named vertices. This relation may be verbally expressed as follows: If a quadrilateral is inscribable in a circle then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides. Moreover, the converse of Ptolemy's theorem is also true: In a quadrilateral, if the sum of the products of the lengths of its two pairs of opposite sides is equal to the product of the lengths of its diagonals, then the quadrilateral can be inscribed in a circle i.e. it is a cyclic quadrilateral. (Wikipedia).
A Beautiful Proof of Ptolemy's Theorem.
Ptolemy's Theorem seems more esoteric than the Pythagorean Theorem, but it's just as cool. In fact, the Pythagorean Theorem follows directly from it. Ptolemy used this theorem in his astronomical work. Google for the historical details. Thanks to this video for the idea of this visual
From playlist Mathy Videos
Ptolemy's theorem and generalizations | Rational Geometry Math Foundations 131 | NJ Wildberger
The other famous classical theorem about cyclic quadrilaterals is due to the great Greek astronomer and mathematician, Claudius Ptolemy. Adopting a rational point of view, we need to rethink this theorem to state it in a purely algebraic way, without resort to `distances' and the correspon
From playlist Math Foundations
Calculus - The Fundamental Theorem, Part 1
The Fundamental Theorem of Calculus. First video in a short series on the topic. The theorem is stated and two simple examples are worked.
From playlist Calculus - The Fundamental Theorem of Calculus
Teach Astronomy - Universal Law of Gravity
http://www.teachastronomy.com/ Newton's master work is the universal law of gravity. Newton's law of gravity states that every object in the universe, every particle, every planet, every star, every galaxy, attracts each other with a force that is proportional to each of the masses of two
From playlist 03. Concepts and History of Astronomy and Physics
Teach Astronomy - Inverse Square Law
http://www.teachastronomy.com/ The key to Newton's realization of the universal law of gravity was the understanding that gravity is an inverse square law. That is, the force of gravity diminishes with the square of the distance between two objects. Using this understanding, Newton was a
From playlist 03. Concepts and History of Astronomy and Physics
A Miraculous Proof (Ptolemy's Theorem) - Numberphile
Featuring Zvezdelina Stankova... Want more? Part 2 (bringing in Pentagons and the Golden Ratio) is at: https://youtu.be/o3QBgkQi_HA More links & stuff in full description below ↓↓↓ Zvezda's Numberphile playlist: http://bit.ly/zvezda_videos Zvezda's webpage: https://math.berkeley.edu/~s
From playlist Women in Mathematics - Numberphile
http://www.teachastronomy.com/ Pythagoras was one of the most influential thinkers in history. This Greek philosopher and mathematician came up with the idea that numbers were the basis of everything. There is no written record, and nothing about Pythagoras survives in writing. He essen
From playlist 02. Ancient Astronomy and Celestial Phenomena
Episode 6: Sines And Cosines Part III - Project MATHEMATICS!
Episode 6. Sines and Cosines, Part III: (Addition formulas) Animation relates the sine and cosine of an angle with chord lengths of a circle, as explained in Ptolemy’s Almagest. This leads to elegant derivations of addition formulas, with applications to simple harmonic motion. A Program
From playlist Courses and Series
Pentagons and the Golden Ratio - Numberphile
Continuing on from Zvezda's previous video about Ptolemey's Theorem (see: https://youtu.be/bJOuzqu3MUQ) now we use it to prove some cool stuff with pentagons and equilateral triangles. More links & stuff in full description below ↓↓↓ Zvezda's Numberphile playlist: http://bit.ly/zvezda_vid
From playlist Women in Mathematics - Numberphile
Teach Astronomy - Newton and Cosmology
http://www.teachastronomy.com/ Newton viewed both time and space as smooth, absolute, and Euclidian. Newton's gravity law is an inverse square law, so the gravity of every object diminishes with the square of the distance. However it never reaches zero because one over the square of a la
From playlist 04. Chemistry and Physics
Boris Springborn: Discrete Uniformization and Ideal Hyperbolic Polyhedra
CATS 2021 Online Seminar Boris Springborn, Technical University of Berlin Abstract: This talk will be about two seemingly unrelated problems: 00:46:00 A discrete version of the uniformization problem for piecewise flat surfaces, and 00:35:48 Constructing ideal hyperbolic polyhedra with p
From playlist Computational & Algorithmic Topology (CATS 2021)
Theory of numbers: Congruences: Euler's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim
From playlist Theory of numbers
Determining if a vector is a linear combination of other vectors
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Determining if a vector is a linear combination of other vectors
From playlist Linear Algebra
We present updates to the automated geometric functionality of the Wolfram Language introduced in Version 12, including new functionality for automated geometric reasoning and for creating GeometricScene objects.
From playlist Wolfram Technology Conference 2022
Maps between Surfaces by Athanase Papadopoulos
PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis
From playlist Ergodic Theory and Dynamical Systems 2022
Setting a New Stage for History of Science - N. Swerdlow - 4/26/2019
On April 26-27 2019, the Division of Humanities & Social Sciences at Caltech hosted a conference in honor of Jed Z. Buchwald, “Looking Back as We Move Forward: The Past, Present, and Future of the History of Science.” This event was sponsored by the Division of the Humanities & Social Sci
From playlist Looking Back as We Move Forward - A Conference in Honor of Jed Z. Buchwald - 4/26-27/2019
Calculus 5.3 The Fundamental Theorem of Calculus
My notes are available at http://asherbroberts.com/ (so you can write along with me). Calculus: Early Transcendentals 8th Edition by James Stewart
From playlist Calculus
Physics 20B. Cosmology. Lec. 2: The Dawn of Science
UCI Physics 20B: Cosmology (Winter 2015) Lec 02. Cosmology -- The Dawn of Science View the complete course: http://ocw.uci.edu/courses/physics_20b_cosmology.html Instructor: James Bullock, Ph.D. License: Creative Commons CC-BY-SA Terms of Use: http://ocw.uci.edu/info More courses at http
From playlist Physics 20B: Cosmology