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Ptolemy's theorem
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===Rectangle=== [[Image:Ptolemy Rectangle.svg|right|thumb|Pythagoras's theorem: ''"manifestum est"'': Copernicus]] More generally, if the quadrilateral is a [[rectangle]] with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d<sup>2</sup>, the right hand side of Ptolemy's relation is the sum ''a''<sup>2</sup> + ''b''<sup>2</sup>. Copernicus β who used Ptolemy's theorem extensively in his trigonometrical work β refers to this result as a 'Porism' or self-evident corollary: :''Furthermore it is clear ('''manifestum est''') that when the chord subtending an arc has been given, that chord too can be found which subtends the rest of the semicircle.''<ref>[http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1543droc.book.....C&db_key=AST&page_ind=36&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES De Revolutionibus Orbium Coelestium: Page 37]. See last two lines of this page. Copernicus refers to Ptolemy's theorem as [http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1543droc.book.....C&db_key=AST&page_ind=37&plate_select=NO&data_type=GIF&type=SCREEN_GIF&classic=YES "Theorema Secundum".]</ref>
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