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Saturday, July 14, 2012

Lemma

Filed under: General,Geometry — Tags: , — m759 @ 2:25 pm

IMAGE- 'Lemma (mathematics)' in Wikipedia

For example—

A letter to the editor of the American Mathematical Monthly
from the June-July 1985 issue has—

… a "square-triangle" lemma:

   ( t ∈ T , t  is an  -replica )
    if and only if  
n  is a square.

  [I.e., "Every triangle is an -replica"
   is true if and only if n  is a square.]

For definitions, see the 1985 letter in Triangles Are Square.

(The 1984 lemma discussed there has now, in response to an article
in Wolfram MathWorld, been renamed the square-triangle theorem .)

A search today for related material yielded the following—

"Suppose that one side of a triangle
has length . Then it can be cut
into n  2 congruent triangles which
are similar to the original one and
whose corresponding sides to the
side of length  have lengths 1."

This was supplied, without attribution, as part of the official solution
to Problem 3 in the 17th Asian Pacific Mathematics Olympiad
from March 2005. Apparently it seemed obvious to the composer
of the problem. As the 1985 letter notes, it may be not quite  obvious.

At any rate, it served in Problem 3 as a lemma , in the sense
described above by Wikipedia. See related remarks by Doron Zeilberger.

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