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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.
Lucas's theorem (number theory) Lukacs's proportion-sum independence theorem (probability) Lumer–Phillips theorem (semigroup theory) Luzin's theorem (real analysis) Lyapunov–Malkin theorem (stability theory) Lyapunov's central limit theorem (probability theory)
Given such a constant k, the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by {(,): =}. A direct proportionality can also be viewed as a linear equation in two variables with a y -intercept of 0 and a slope of k > 0, which corresponds to linear growth .
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Similar triangles provide the basis for many synthetic (without the use of coordinates) proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem.
Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. [3] The model theoretic notion has the advantage over the proof theoretic one that it does not depend so much on the language at hand; on the other hand, it is usually harder to establish model theoretic conservativity.
An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis . For some time it was thought that certain theorems, like the prime number theorem , could only be proved using "higher" mathematics.
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 ...