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Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle.. 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.
Castelnuovo–de Franchis theorem (algebraic geometry) Chow's theorem (algebraic geometry) Cramer's theorem (algebraic curves) (analytic geometry) Hartogs's theorem (complex analysis) Hartogs's extension theorem (several complex variables) Hirzebruch–Riemann–Roch theorem (complex manifolds) Kawamata–Viehweg vanishing theorem (algebraic ...
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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 .
The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the maximum norm and R m ...
To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
The last step of the proof fails if the projective space has dimension less than 3, as in this case it is not possible to find a point not in the plane. Monge's theorem also asserts that three points lie on a line, and has a proof using the same idea of considering it in three rather than two dimensions and writing the line as an intersection ...
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. [69] The term was introduced by János Bolyai in 1832. [70] It is sometimes referred to as neutral geometry, [71] as it is neutral with respect to the parallel postulate.
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