By Elizabeth J Billington; Sheila Oates-Williams; Anne Penfold Street
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Additional resources for Combinatorial mathematics IX : proceedings of the Ninth Australian Conference on Combinatorial Mathematics, held at the University of Queensland, Brisbane, Australia, August 24-28, 1981
28: Annotated Perigal Diagram We are going to leave the proof to the reader as a challenge. Central to the Perigal argument is the fact that all eight of the constructed quadrilaterals are congruent. This immediately leads to fact that the middle square embedded in the ‘c’ square is identical to the ‘a’ square from which a 2 " b 2 ! c 2 can be established. Perigal’s proof has since been cited as one of the most ingenious examples of a proof associated with a phenomenon that modern mathematicians call a Pythagorean Tiling.
12. Euclid finally used a formal verballydescriptive logic stream quite similar to the annotated algebraic logic stream below in order to complete his proof. : A 2 " B 2 ! : B ! :3B ' A ! : A2 " ( B ' ) 2 ! : X 2 ! A 2 " ( B ' ) 2 & X 2 ! A2 " B 2 & X 2 ! :$ ! :" " # " $ ! 180 0 & " " # " 90 0 ! 180 0 & " " # ! $ ( 8] Properties of algebraic equality We close this section by simply admiring the simple and profound algebraic symmetry of the Pythagorean Theorem and its converse as ‘chiseled’ below C # $ " A B " " # !
2 set ! : (a " b) 2 ! c 2 " 4( 12 ab) & a 2 " 2ab " b 2 ! c 2 " 2ab & a2 " b2 ! c2 ( 31 Notice how quickly and easily our result is obtained once algebraic is used to augment the geometric picture. Simply put, algebra coupled with geometry is superior to geometry alone in quantifying and tracking the diverse and subtle relationships between geometric whole and the assorted pieces. Hence, throughout the remainder of the book, analytic geometry will be used to help prove and develop results as much as possible.
Combinatorial mathematics IX : proceedings of the Ninth Australian Conference on Combinatorial Mathematics, held at the University of Queensland, Brisbane, Australia, August 24-28, 1981 by Elizabeth J Billington; Sheila Oates-Williams; Anne Penfold Street