Since A lies on M, so does B, and the circle M is therefore the triangle's circumcircle. This means that A and B are equidistant from the origin, i.e. Since the sum of the angles of a triangle is equal to 180°, we have The three internal angles of the ∆ ABC triangle are α, ( α + β), and β. Since OA = OB = OC, △ OBA and △ OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠ OBC = ∠ OCB and ∠ OBA = ∠ OAB. The following facts are used: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal. 300 BC) as proposition 31: "In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle."ĭante's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech. The theorem appears in Book III of Euclid's Elements ( c. Modern scholars believe that Greek deductive geometry as found in Euclid's Elements was not developed until the 4th century BC, and any geometric knowledge Thales may have had would have been observational. However, there is no direct evidence for any of these claims, and they were most likely invented speculative rationalizations. Thales was claimed to have traveled to Egypt and Babylonia, where he is supposed to have learned about geometry and astronomy and thence brought their knowledge to the Greeks, along the way inventing the concept of geometric proof and proving various geometric theorems. Reference to Thales was made by Proclus (5th century AD), and by Diogenes Laërtius (3rd century AD) documenting Pamphila's (1st century AD) statement that Thales "was the first to inscribe in a circle a right-angle triangle". Thales of Miletus (early 6th century BC) is traditionally credited with proving the theorem however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxographers based on hearsay and speculation. – English translation by Henry Wadsworth Longfellowīabylonian mathematicians knew this for special cases before Greek mathematicians proved it. – Dante's Paradiso, Canto 13, lines 100–102 It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle. For the theorem sometimes called Thales' theorem and pertaining to similar triangles, see intercept theorem.
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