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Proof without words of the Nicomachus theorem (Gulley (2010)) that the sum of the first n cubes is the square of the n th triangular number. In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
Cheng's eigenvalue comparison theorem (Riemannian geometry) Chern–Gauss–Bonnet theorem (differential geometry) Classification of symmetric spaces ; Darboux's theorem (symplectic topology) Euler's theorem (differential geometry) Four-vertex theorem (differential geometry) Frobenius theorem ; Gauss's lemma (riemannian geometry)
Other common auxiliary constructs in elementary plane synthetic geometry are the helping circles. As an example, a proof of the theorem on the sum of angles of a triangle can be done by adding a straight line parallel to one of the triangle sides (passing through the opposite vertex).
Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs: No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one.
In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve).There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
Printable version; In other projects Wikidata item; Appearance. move to sidebar hide. This is a short list of some common mathematical shapes and figures and the ...
In 2015, an anonymous Japanese woman using the pen name "aerile re" published the first known method (the method of 3 circumcenters) to construct a proof in elementary geometry for a special class of adventitious quadrangles problem. [7] [8] [9] This work solves the first of the three unsolved problems listed by Rigby in his 1978 paper. [5]
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