Editing Nine-point circle (section)

{{Short description|Circle constructed from a triangle}}
[[File:Triangle.NinePointCircle.svg|200px|thumb|The nine points]]

<!-- [[File:EulerCircle3.gif|thumb|You can change the vertices of the triangle, and Euler's circle persists.]] -->In [[geometry]], the '''nine-point circle''' is a [[circle]] that can be constructed for any given [[triangle]]. It is so named because it passes through nine significant [[concyclic points]] defined from the triangle. These nine [[point (geometry)|points]] are:

* The [[midpoint]] of each side of the triangle
* The [[Perpendicular|foot]] of each [[altitude (triangle)|altitude]]
* The midpoint of the [[line segment]] from each [[vertex (geometry)|vertex]] of the triangle to the [[orthocenter]] (where the three altitudes meet; these line segments lie on their respective altitudes).<ref>{{harvtxt|Altshiller-Court|1925|pp=103–110}}</ref><ref>{{harvtxt|Kay|1969|pp=18,245}}</ref>

The nine-point circle is also known as  '''Feuerbach's circle''' (after [[Karl Wilhelm Feuerbach]]), '''Euler's circle''' (after [[Leonhard Euler]]),  '''Terquem's circle''' (after [[Olry Terquem]]), the '''six-points circle''', the '''twelve-points circle''', the '''{{mvar|n}}-point circle''', the '''medioscribed circle''', the '''mid circle''' or the '''circum-midcircle'''. Its center is the [[nine-point center]] of the triangle.<ref>{{cite journal|author=Kocik, Jerzy|author2=Solecki, Andrzej|title=Disentangling a Triangle|journal=Amer. Math. Monthly|volume=116|issue=3|year=2009|pages=228–237|url=http://www.maa.org/programs/maa-awards/writing-awards/disentangling-a-triangle|doi=10.4169/193009709x470065}} Kocik and Solecki (sharers of a 2010 [[Lester R. Ford Award]]) give a proof of the Nine-Point Circle Theorem.</ref><ref>{{cite book|author=Casey, John|author-link=John Casey (mathematician)|title=''Nine-Point Circle Theorem, in'' A Sequel to the First Six Books of Euclid|page=58|year=1886|edition=4th|location=London|publisher=Longmans, Green, & Co|url=http://babel.hathitrust.org/cgi/pt?id=hvd.hn6mqv;view=1up;seq=78}}</ref>