Editing Ptolemy's theorem (section)

===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>