Thales' theorem

Thales' theorem states that an angle inscribed in a semicircle is a right angle. Equivalently, a triangle whose longest side is a circle's diameter is right-angled.

Inscribed angle theorem: An inscribed angle measures half the central angle subtending the same arc. The theorem follows when the corresponding central angle spans a semicircle.
Chord (geometry): A line segment whose endpoints lie on a circle. The two sides meeting at the semicircle's point are chords.
Thales' circle: The circle having a given segment as its diameter. Every point on this circle forms a right angle with the segment's endpoints.
Thales of Miletus: A Greek philosopher and mathematician traditionally dated to the sixth century BCE. The theorem bears Thales' name, though the historical attribution is uncertain.
Circle: A plane curve whose points are all equally distant from a fixed center. The circle supplies the arc on which the right angle's vertex lies.
Semicircle: Half of a circle bounded by a diameter and a 180-degree arc. The theorem specifically places the angle's vertex on this half-circle's arc.
Geometric construction: A method for constructing geometric figures with specified tools and constraints. A diameter and its circle construct points that yield right angles.
Ancient Greek mathematics: Mathematical inquiry developed in the Greek-speaking world from antiquity onward. The theorem is traditionally associated with early Greek geometric proof.
Diameter: A line segment through a circle's center with endpoints on its circumference. Its endpoints define the semicircle and form the triangle's hypotenuse.
Hypotenuse: The side opposite the right angle in a right triangle. The diameter becomes the hypotenuse of the triangle in the theorem.

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