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Descartes's theorem (plane geometry) Dinostratus' theorem (geometry, analysis) Equal incircles theorem (Euclidean geometry) Euler's quadrilateral theorem ; Euler's theorem in geometry (triangle geometry) Exterior angle theorem (triangle geometry) Feuerbach's theorem ; Finsler–Hadwiger theorem ; Five circles theorem
The second theorem considers five circles in general position passing through a single point M. Each subset of four circles defines a new point P according to the first theorem. Then these five points all lie on a single circle C. The third theorem considers six circles in general position that pass through a single point M. Each subset of five ...
Brianchon's theorem can be proved by the idea of radical axis or reciprocation. To prove it take an arbitrary length (MN) and carry it on the tangents starting from the contact points: PL = RJ = QH = MN etc. Draw circles a, b, c tangent to opposite sides of the hexagon at the created points (H,W), (J,V) and (L,Y) respectively.
Kempe's proof did, however, suffice to show the weaker five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. [2] Schröder–Bernstein theorem. In 1896 Schröder published a proof sketch [3] which, however, was shown to be faulty by Alwin Reinhold Korselt in 1911 [4] (confirmed by ...
Ceva's theorem can be obtained from it by setting the area equal to zero and solving. The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century. [9] The theorem has also been generalized to triangles on other surfaces of constant curvature. [10]
The theorem is "remarkable" because the definition of Gaussian curvature makes ample reference to the specific way the surface is embedded in 3-dimensional space, and it is quite surprising that the result does not depend on its embedding. In modern mathematical terminology, the theorem may be stated as follows:
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Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): [16] "Let the following be postulated": "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line."