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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
[7]: xii–xiii, xvii–xviii The section on geometry (IX) contains many problems contributed by Loewner (in differential geometry) and Hirsch (in algebraic geometry). [4]: 27 The book was unique at the time because of its arrangement, less by topic and more by method of solution, so arranged in order to build up the student's problem-solving ...
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 ...
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:
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 ...