Editing Ptolemy's theorem (section)

== Corollaries on inscribed polygons ==
===Equilateral triangle===
[[Image:Ptolemy Equilateral.svg|right|thumb|Equilateral triangle]]

Ptolemy's Theorem yields as a corollary a theorem<ref name="wilson">[http://jwilson.coe.uga.edu/emt725/Ptolemy/Ptolemy.html Wilson, Jim. "Ptolemy's Theorem."] link verified 2009-04-08</ref> regarding an equilateral triangle inscribed in a circle.

'''Given''' An equilateral triangle inscribed on a circle, and a point on the circle.

The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.

'''Proof:''' Follows immediately from Ptolemy's theorem:

:<math> qs=ps+rs \Rightarrow q=p+r. </math>

This corollary has as an application an algorithm for computing minimal [[Steiner tree]]s whose topology is fixed, by repeatedly replacing pairs of leaves of the tree ''A'', ''B'' that should be connected to a [[Steiner point (computational geometry)|Steiner point]], by the third point ''C'' of their equilateral triangle. The unknown Steiner point must lie on arc ''AB'' of the circle, and this replacement ensures that, no matter where it is placed, the length of the tree remains unchanged.<ref>{{citation|title=The Shortest-Network Problem|first1=Marshall W.|last1=Bern|first2=Ronald L.|last2=Graham|author2-link=Ronald Graham|journal=[[Scientific American]]|volume=260|issue=1|date=January 1989|pages=84–89|doi=10.1038/scientificamerican0189-84 |jstor=24987111|bibcode=1989SciAm.260a..84B |url=https://mathweb.ucsd.edu/~ronspubs/89_01_shortest_network.pdf}}</ref>

===Square===

Any [[Square (geometry)|square]] can be inscribed in a circle whose center is the center of the square. If the common length of its four sides is equal to <math>a</math> then the length of the diagonal is equal to <math>a\sqrt{2}</math> according to the [[Pythagorean theorem]], and Ptolemy's relation obviously holds.

===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>&nbsp;+&nbsp;''b''<sup>2</sup>.

Copernicus – who used Ptolemy's theorem extensively in his trigonometrical work – 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>

===Pentagon===
[[Image:Ptolemy Pentagon.svg|right|thumb|The [[golden ratio]] follows from this application of Ptolemy's theorem]]
A more interesting example is the relation between the length ''a'' of the side and the (common) length ''b'' of the 5 chords in a regular pentagon. By [[completing the square]], the relation yields the [[golden ratio]]:<ref>[http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII8.html Proposition 8] in Book XIII of [http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html Euclid's Elements] proves by similar triangles the same result: namely that length a (the side of the pentagon) divides length b (joining alternate vertices of the pentagon) in "mean and extreme ratio".</ref>
:<math>\begin{array}{rl}
         b \cdot b                  \,\;\;\qquad\quad\qquad =&\!\!\!\! a \! \cdot \! a + a \! \cdot \! b
\\       b^2  \;\;  - ab                        \quad\qquad =&\!\!     a^2
\\ \frac{b^2}{a^2} \;\; - \frac{ab}{a^2}       \;\;\;\qquad =&\!\!\!        \frac{        a^2 }{a^2}
\\ \left(\frac{b}{a}\right)^2 - \frac{b}{a} + \left(\frac{1}{2}\right)^2 =&\!\!     1 + \left(\frac{          1 }{  2}\right)^2
\\ \left(\frac{b}{a}         -                  \frac{1}{2}\right)^2     =&\!\!    \quad \frac{          5 }{  4}
\\  \frac{b}{a}         -                \frac{1}{2} \;\;\; =&\!\!\!\!  \pm \frac{    \sqrt{5}}{  2}
\\  \frac{b}{a} > 0 \, \Rightarrow \, \varphi = \frac{b}{a} =&\!\!\!\!      \frac{1 + \sqrt{5}}{  2}
\end{array}</math>

===Side of decagon===
[[Image:Ptolemy Pentagon2.svg|right|thumb|Side of the inscribed decagon]]
If now diameter AF is drawn bisecting DC so that DF and CF are sides c of an inscribed decagon, Ptolemy's Theorem can again be applied – this time to cyclic quadrilateral ADFC with diameter ''d'' as one of its diagonals:

:<math>ad=2bc</math>

:<math>\Rightarrow ad=2\varphi ac</math> where <math>\varphi</math> is the golden ratio.
:<math>\Rightarrow c=\frac{d}{2\varphi}.</math><ref>And in analogous fashion [http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII9.html Proposition 9] in Book XIII of [http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html Euclid's Elements] proves by similar triangles that length c (the side of the decagon) divides the radius in "mean and extreme ratio".</ref>

whence the side of the inscribed decagon is obtained in terms of the circle diameter. Pythagoras's theorem applied to right triangle AFD then yields "b" in terms of the diameter and "a" the side of the pentagon <ref>An interesting article on the construction of a regular pentagon and determination of side length can be found at the following reference [http://www.cut-the-knot.org/pythagoras/pentagon.shtml]</ref> is thereafter calculated as
::<math>a = \frac {b} {\varphi} = b \left( \varphi - 1 \right).</math>

As [[Copernicus]] (following Ptolemy) wrote,

:''"The diameter of a circle being given, the sides of the triangle, tetragon, pentagon, hexagon and decagon, which the same circle circumscribes, are also given."''<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: Liber Primus: Theorema Primum]</ref>