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Thought following might be of interest - application of Ptolemy's Theorem from "De Revolutionibus Orbium Coelestium: Liber Primus".
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1. Construct a unit diameter from either end of the chord. Complete right triangle by joining end of diameter to other end of chord.dextercioby said:Amazing visual proof of the trigonometric identity, but 2 things are not clear to me.
1. Why is the upper chord ##\sin(\beta-\alpha)##?
2. What would a purely geometrical (i.e. trigonometry-free) proof of the equality "chord times diameter equals diagonals' product minus the product of the other chords" be?
Ptolemy's Theorem is a mathematical concept developed by the ancient Greek astronomer and mathematician, Claudius Ptolemy. It states that in a cyclic quadrilateral (a four-sided figure inscribed within a circle), the product of the diagonals is equal to the sum of the products of the opposite sides.
Ptolemy's Theorem was first introduced in Ptolemy's book, "Almagest," which was published around 150 AD.
Ptolemy's Theorem has been used in various fields, including astronomy, geometry, and engineering. It has also been applied in solving complex trigonometric equations and determining the position of celestial bodies.
Ptolemy's Theorem was used by ancient civilizations, such as the Greeks and Egyptians, to accurately predict the movements of the planets and stars. It also played a crucial role in the development of trigonometry and navigation.
Yes, Ptolemy's Theorem is still used in modern mathematics and engineering. It is also a fundamental concept taught in geometry and trigonometry courses.