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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.
For fixed points A and B, the set of points M in the plane for which the angle ∠AMB is equal to α is an arc of a circle. The measure of ∠ AOB , where O is the center of the circle, is 2 α . The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2 θ that intercepts the same arc on the circle.
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 ]
The latter two can be done with a construction based on the intercept theorem. A slightly less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length from a constructed segment of length . It follows that every algebraically constructible number is geometrically constructible, by ...
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
Elitzur's theorem (quantum field theory, statistical field theory) Envelope theorem (calculus of variations) Equal incircles theorem (Euclidean geometry) Equidistribution theorem (ergodic theory) Equipartition theorem (ergodic theory) Erdős–Anning theorem (discrete geometry) Erdős–Dushnik–Miller theorem ; Erdős–Gallai theorem (graph ...
The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio. [1] [2] The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.
Cardinality – Definition of the number of elements in a set; Complement – Set of the elements not in a given subset; Intersection (Euclidean geometry) – Shape formed from points common to other shapes; Intersection graph – Graph representing intersections between given sets