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In geometry, Thales' theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Geometry (Greek γεωμετÏία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...
Thales Thales (in Greek: Θαλής) of Miletus (ca. ...
In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). ...
This article is about angles in geometry. ...
This article is about angles in geometry. ...
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Proof
We use the following facts: the sum of the angles in a triangle is equal to two right angles and that the base angles of an isosceles triangle are equal. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
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Let O be the center of the circle. Since OA = OB = OC, OAB and OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, OBC = OCB and BAO = ABO. Let γ = BAO and δ = OBC.
Since the sum of the angles of a triangle is equal to two right angles, we have
- 2γ + γ ′ = 180°
and - 2δ + δ ′ = 180°
We also know that - γ ′ + δ ′ = 180°
Adding the first two equations and subtracting the third, we obtain - 2γ + γ ′ + 2δ + δ ′ − (γ ′ + δ ′) = 180°
which, after cancelling γ ′ and δ ′, implies that - γ + δ = 90°
Q.E.D. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, which was to be demonstrated). This is a translation of the Greek (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. ...
Converse The converse of Thales' theorem is also valid, which states that a right triangle's hypotenuse is a diameter of its circumcircle. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ...
The theorem and its converse can be expressed as follows:
- The center of the circumcircle of a triangle lies on one of the triangle's sides if and only if the triangle is a right triangle.
↔ ⇔ ≡ logical symbols representing iff. ...
Proof of the converse The proof utilises the fact that directional vectors of two lines form right angles if and only if the dot product is zero. Let there be a right angle ABC and circle M with AC as a diameter. Let M's center lie on the origin, for easier calculation. Then the dot product of AB and BC is In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
- (A − B) · (B − C) = (A − B) · (B + A) = |A|2 − |B|2 = 0.
Hence: - |A| = |B|.
A and B are also equidistant from the circle's center, hence M is the triangle's circumcenter.
Generalization Thales' theorem is a special case of the following theorem: given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.
History Thales was not the first to discover this theorem since the Egyptians and Babylonians must have known of this empirically. However they did not prove the theorem, and the theorem is named after Thales because he was said to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to two right angles. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
External link - Munching on Inscribed Angles
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