Editing Lemniscate of Bernoulli (section)

==Angles==
Given two distinct points <math>\rm A</math> and <math>\rm B</math>, let <math>\rm M</math> be the midpoint of <math>\rm AB</math>. Then the lemniscate of [[Diameter of a set|diameter]] <math>\rm AB</math> can also be defined as the set of points <math>\rm A</math>, <math>\rm B</math>, <math>\rm M</math>, together with the locus of the points <math>\rm P</math> such that <math>|\widehat{\rm APM}-\widehat{\rm BPM}|</math> is a right angle (cf. [[Thales' theorem]] and its converse).<ref>{{Cite book |last1=Eymard |first1=Pierre |last2=Lafon| first2=Jean-Pierre |title=The Number Pi |publisher=American Mathematical Society |year=2004 |isbn=0-8218-3246-8}} p. 200</ref>

[[File:Lemniskate vechtmann.svg|thumb|upright=1.75|relation between angles at Bernoulli's lemniscate]]
The following theorem about angles occurring in the lemniscate is due to German mathematician [[Gerhard Christoph Hermann Vechtmann]], who described it 1843 in his dissertation on lemniscates.<ref>Alexander Ostermann, Gerhard Wanner: ''Geometry by Its History.'' Springer, 2012, pp. [https://books.google.com/books?id=eOSqPHwWJX8C&pg=PA207 207-208]</ref> 

:{{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}} are the foci of the lemniscate, {{math|''O''}} is the midpoint of the line segment {{math|''F''<sub>1</sub>''F''<sub>2</sub>}} and {{math|''P''}} is any point on the lemniscate outside the line connecting {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}}. The normal {{math|''n''}} of the lemniscate in {{math|''P''}} intersects the line connecting {{math|''F''<sub>1</sub>}} and {{math|''F''<sub>2</sub>}} in {{math|''R''}}. Now the interior angle of the triangle {{math|''OPR''}} at {{math|''O''}} is one third of the triangle's exterior angle at {{math|''R''}} (see also [[angle trisection]]). In addition the interior angle at {{math|''P''}} is twice the interior angle at {{math|''O''}}.