It can be argued that triangles and the Pythagorean Theorem are prime movers in the development of western thought. Many modern theorems contain concepts related to triangles buried within them. One of the most fundamental relationships in triangles is the Pythagorean Theorem of triangles containing a right angle. The oldest record of the “Pythagorean” Theorem comes from Babylon (Plimpton 322 clay tablet dated to 1800 BC), and attribution of the theorem to the legendary Greek mathematician Pythagoras is erroneous. Regardless, the Pythagorean Theorem along with Euclid’s Elements and other Theorems from the classical Greek age (e.g., Thales Inscribed Angle Theorem, Pappus Centroid Theorem, etc.) don’t just describe methods of understanding 2D & 3D space, these theorems are the basis of a significant amount of modern math. Further, the development of proofs in the field of geometry inspired people to look for methods of proof in other areas ranging from empirical science to law.
There are more than 100 good proofs of the Pythagorean Theorem (see “The Pythagorean Proposition”; Elisha S. Loomis; Classics in Mathematics Education; copyright 1940). President Andrew Garfield published an interesting proof of the Pythagorean Theorem in 1876 while he was still a congressman. Perhaps the most easily recognized proof is given in the graphic below.
Note that many simple trigonometric relations are easily derived from the Pythagorean Theorem (see the graphic below).
Note that the Pythagorean Theorem and the Parallel Line Postulate of Euclid are all that is necessary to prove the famous quadratic mean, arithmetic mean, geometric mean, harmonic mean inequality in statistics:
QM > AM > GM > HM
The elucidation of basic relationships in triangles led to the beginning of logical proofs in general. See the paper and excel files linked below. The paper is a ramble through many basic and derived relationships in triangles along with a few new ideas. One Excel sheet is a bit busy but provides a means of producing most of the graphics contained in the paper. The ten circles Excel sheet produces a graphic picture of 10 circles inside a right triangle (you pick the leg lengths).
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