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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'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 . [ 1 ]
A special case of the theorem is Thales's theorem, which states that the angle subtended by a diameter is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.
Thales' theorem implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. [21] If the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. [22]
It was published in Oxford in 1921, in two volumes titled Volume I, From Thales to Euclid and Volume II, From Aristarchus to Diophantus. It got positive reviews and is still used today. Ten years later, in 1931, Heath published A Manual of Greek Mathematics, a concise version of the two-volume History.
Diagram to show the converse to Thales' theorem. This is that given a right-angle, ABC, the circle centred on the midpoint of AC, O, passes through B. This is that given a right-angle, ABC, the circle centred on the midpoint of AC, O, passes through B.
Thales's theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle. Pamphila says that, having learnt geometry from the Egyptians, Thales was the first to inscribe in a circle a right-angled triangle, whereupon he sacrificed an ox . [ 54 ]