We show how the basic notion of rational trigonometry---quadrance---arises from the geometry of the ancient Greeks. The little-known sister theorem to Pythagoras features prominently, and is closely related to a theorem of Archimedes.
For someone brought up on classical trigonometry, quadrance is the square of the distance. But future generations will come around to the Euclidean realization that it is quadrance which is actually the fundamental quantity: distance is the square root of quadrance--assuming (and it is a BIG assumption) that you have a viable theory of square roots around.
Pythagoras' theorem is matched up with the Triple quad formula (TQF) ---in my opinion these are the two most important theorems in mathematics. The reach of the TQF is long indeed.
Video Contents:
00:00 Points, lines and circles and the role of quadrance
03:16 Proof of Pythagoras theorem
04:51 Lemma ( Proof of Triple quad formula)
08:00 A symmetrical version
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Screenshot PDFs for my videos are available at the website [ Ссылка ]. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at [ Ссылка ]
My blog is at [ Ссылка ], where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at [ Ссылка ] Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at [ Ссылка ] Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
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Here are the Wild Egg Maths Playlists (some available only to Members!)
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