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The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles , see Trigonometric functions . Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine , or on the differential equation f ″ + f = 0 ...
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
Menelaus's theorem, case 1: line DEF passes inside triangle ABC. In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A ...
As stated above, Thales's theorem is a special case of the inscribed angle theorem (the proof of which is quite similar to the first proof of Thales's theorem given above): Given three points A, B and C on a circle with center O, the angle ∠ AOC is twice as large as the angle ∠ ABC. A related result to Thales's theorem is the following:
Steiner–Lehmus theorem (triangle geometry) Steinhaus theorem (measure theory) Steinitz theorem (graph theory) Stewart's theorem (plane geometry) Stinespring factorization theorem (operator theory) Stirling's theorem (mathematical analysis) Stokes's theorem (vector calculus, differential topology) Stolper–Samuelson theorem
The following sample of theorems gives a flavor of the new results discovered by Discoverer. Theorem 6.1 Let P and Q are points, neither lying on a sideline of triangle ABC. If P and Q are isogonal conjugates with respect to ABC, then the Ceva product of their complements lies on the Kiepert hyperbola. Theorem 9.1.
Diagram of Stewart's theorem. Let a, b, c be the lengths of the sides of a triangle. Let d be the length of a cevian to the side of length a.If the cevian divides the side of length a into two segments of length m and n, with m adjacent to c and n adjacent to b, then Stewart's theorem states that + = (+).
It can be shown that two triangles having congruent angles (equiangular triangles) are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. [2] Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle".