https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=BeapBeap - Revision history2026-08-10T18:14:03ZRevision history for this page on the wikiMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Beap&diff=245179&oldid=preven>Jochen Burghardt: /* References */ proper citation from heapsort article2014-11-26T18:03:20Z<p><span class="autocomment">References: </span> proper citation from <a href="/wiki/Heapsort" title="Heapsort">heapsort</a> article</p>
<a href="https://en.formulasearchengine.com/w/index.php?title=Beap&diff=245179&oldid=12018">Show changes</a>en>Jochen Burghardthttps://en.formulasearchengine.com/w/index.php?title=Beap&diff=12018&oldid=preven>Cydebot: Robot - Moving category Heaps (structure) to :Category:Heaps (data structures) per CFD at Wikipedia:Categories for discussion/Log/2012 January 12.2012-01-17T01:07:08Z<p>Robot - Moving category Heaps (structure) to <a href="/w/index.php?title=Category:Heaps_(data_structures)&action=edit&redlink=1" class="new" title="Category:Heaps (data structures) (page does not exist)">Category:Heaps (data structures)</a> per <a href="/w/index.php?title=WP:CFD&action=edit&redlink=1" class="new" title="WP:CFD (page does not exist)">CFD</a> at <a href="https://en.wikipedia.org/wiki/Categories_for_discussion/Log/2012_January_12" class="extiw" title="wikipedia:Categories for discussion/Log/2012 January 12">Wikipedia:Categories for discussion/Log/2012 January 12</a>.</p>
<p><b>New page</b></p><div>In [[geometry]], '''Euler's theorem''' states that the distance ''d'' between the [[circumcentre]] and [[incentre]] of a [[triangle]] can be expressed as <br />
<br />
:<math> d^2=R (R-2r) \,</math><br />
<br />
where ''R'' and ''r'' denote the circumradius and inradius respectively (the radii of the above two circles). The theorem is named for [[Leonhard Euler]], who published it in 1767. However, the same result was published earlier by William Chapple in 1746.<br />
<br />
From the theorem follows the '''Euler inequality''':<br />
:<math>R \ge 2r. </math><br />
<br />
==Proof==<br />
[[Image:GeometryEulerTheorem.png|300px|thumb|A figure for following the proof (which also contains the<br />
proof here). Made in [[GeoGebra]] software.]]<br />
Let ''O'' be the circumcentre of triangle ''ABC'', and ''I'' be its incentre, the extension of ''AI'' intersects the circumcircle at ''L'', then ''L'' is the midpoint of arc ''BC''. Join ''LO'' and extend it so that it intersects the circumcircle at ''M''. From ''I'' construct a perpendicular to AB, and let D be its foot, then ''ID'' = ''r''. It is not difficult to prove that triangle ''ADI'' is similar to triangle ''MBL'', so ''ID'' / ''BL'' = ''AI'' / ''ML'', i.e. ''ID'' × ''ML'' = ''AI'' × ''BL''. Therefore 2''Rr'' = ''AI'' × ''BL''. Join ''BI'', because<br />
<br />
: angle ''BIL'' = angle ''A'' / 2 + angle ''ABC'' / 2,<br />
<br />
: angle ''IBL'' = angle ''ABC'' / 2 + angle ''CBL'' = angle ''ABC'' / 2 + angle ''A'' / 2,<br />
<br />
therefore angle ''BIL'' = angle ''IBL'', so ''BL'' = ''IL'', and ''AI'' × ''IL'' = 2''Rr''. Extend ''OI'' so that it intersects the circumcircle at ''P'' and ''Q'', then ''PI'' × ''QI'' = ''AI'' × ''IL'' = 2''Rr'', so (''R''&nbsp;+&nbsp;''d'')(''R''&nbsp;&minus;&nbsp;''d'') = 2''Rr'', i.e. ''d''<sup>2</sup> = ''R''(''R''&nbsp;&minus;&nbsp;2''r'').<br />
<br />
==See also==<br />
*[[Bicentric quadrilateral#Fuss' theorem and Carlitz' identity]] for the relation among the same three variables in bicentric quadrilaterals<br />
<br />
==References==<br />
*{{citation<br />
| last1 = Alsina | first1 = Claudi<br />
| last2 = Nelsen | first2 = Roger<br />
| isbn = 9780883853429<br />
| page = 56<br />
| publisher = Mathematical Association of America<br />
| series = Dolciani Mathematical Expositions<br />
| title = When Less is More: Visualizing Basic Inequalities<br />
| url = http://books.google.com/books?id=U1ovBsSRNscC&pg=PA56<br />
| volume = 36<br />
| year = 2009}}.<br />
*{{citation<br />
| last = Chapple | first = William<br />
| journal = Miscellanea Curiosa Mathematica<br />
| pages = 117–124<br />
| title = An essay on the properties of triangles inscribed in and circumscribed about two given circles<br />
| url = http://books.google.com/books?id=a95JAAAAMAAJ&pg=PA118-IA1<br />
| volume = 4<br />
| year = 1746}}. The formula for the distance is near the bottom of p.123.<br />
*{{citation<br />
| last = Debnath | first = Lokenath<br />
| isbn = 9781848165250<br />
| page = 124<br />
| publisher = World Scientific<br />
| title = The Legacy of Leonhard Euler: A Tricentennial Tribute<br />
| url = http://books.google.com/books?id=K2liU-SHl6EC&pg=PA124<br />
| year = 2010}}.<br />
*{{citation<br />
| last = Dunham | first = William<br />
| isbn = 9780883855584<br />
| page = 300<br />
| publisher = Mathematical Association of America<br />
| series = Spectrum Series<br />
| title = The Genius of Euler: Reflections on his Life and Work<br />
| url = http://books.google.com/books?id=M4-zUnrSxNoC&pg=PA300<br />
| volume = 2<br />
| year = 2007}}.<br />
*{{citation<br />
| last = Euler | first = Leonhard | author-link = Leonhard Euler<br />
| journal = Novi Commentarii academiae scientiarum Petropolitanae<br />
| language = Latin<br />
| pages = 103–123<br />
| title = Solutio facilis problematum quorumdam geometricorum difficillimorum<br />
| url = http://www.math.dartmouth.edu/~euler/docs/originals/E325.pdf<br />
| volume = 11<br />
| year = 1767}}.<br />
<br />
==External links==<br />
* [http://mathworld.wolfram.com/EulerTriangleFormula.html Euler's theorem on MathWorld]<br />
<br />
[[Category:Triangle geometry]]<br />
[[Category:Articles containing proofs]]<br />
[[Category:Geometric inequalities]]<br />
<br />
{{Elementary-geometry-stub}}</div>en>Cydebot