Proofs without words

This should really be a comment on Marco Radeschi's answer from Feb 22 involving the area formula for spherical triangles, but since I'm new here I don't have the reputation to leave comments yet. In reply to Igor's comment (on Marco's answer) wondering about an analogous proof for the area formula of hyperbolic triangles: there is one along similar lines, and you're rescued from non-compactness by the fact that asymptotic triangles have finite area. In particular, the proof in the spherical case relies on the fact that the area of a double wedge with angle α is proportional to α; in the hyperbolic case, you need to replace the double wedge with a doubly asymptotic triangle (one vertex in the hyperbolic plane and two vertices on the ideal boundary) and show that if the angle at the finite vertex is α, then the area is proportional to π−α. (That picture is slightly modified from p. 221 of this book, which has the whole proof in more detail.)

50 problèmes Depuis sa publication en février 2005 « 50 PROBLEMES (et plus si affinités) pour les élèves de quatrième et troisième» a connu une diffusion constante qui a justifié sa réédition en septembre 2011. L'intérêt porté à ce recueil d'exercices ne se démentant pas, il nous a semblé pertinent de lui donner ce complément en ligne pouvant guider les choix des enseignants. Il comporte une double classification des problèmes par contenus mathématiques et par type de recherche ainsi qu'un un certain nombre de corrigés et de commentaires pédagogiques ou didactiques, accessibles par le numéro d'ordre du problème dans le livret. Dans la grande tradition de l'IREM de Lyon, les formulations des exercices étant « ouvertes » pour ne pas induire de méthode de résolution, il aurait été paradoxal de donner des corrigés « fermés ». Aussi avons-nous souvent donné plusieurs pistes de recherches possibles pour les élèves faisant la part belle à l'exploration, aux essais, à la prise d'initiative.

Proofs without words From AoPSWiki The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery. Summations The sum of the first odd natural numbers is positive integers is The alternating sum of the first . Nichomauss' Theorem: can be written as the sum of consecutive integers, and consequently that Here, we use the same re-arrangement as the first proof on this page (the sum of first odd integers is a square). This also suggests the following alternative proof: An animated version of this proof can be found in this gallery. The th pentagonal number is the sum of and three times the th triangular number. denotes the th pentagonal number, then The identity , where is the th Fibonacci number. Back to Top Geometric series The infinite geometric series Another proof of the identity The arithmetic-geometric series , also known as Gabriel's staircase.[2] Geometry The Pythagorean Theorem (first of many proofs): the left diagram shows that The area of a triangle is given by is given by for with