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Nagel point
In geometry, the Nagel point is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. Given a triangle ''ABC'', let ''T''''A'', ''T''''B'', and ''T''''C'' be the extouch points in which the ''A''-excircle meets line ''BC'', the ''B''-excircle meets line ''CA'', and ''C''-excircle meets line ''AB'', respectively. The lines ''AT''''A'', ''BT''''B'', ''CT''''C'' concur in the Nagel point ''N'' of triangle ''ABC''. The Nagel point is named after Christian Heinrich von Nagel, a nineteenth-century German mathematician, who wrote about it in 1836.
Another construction of the point ''T''''A'' is to start at ''A'' and trace around triangle ''ABC'' half its perimeter, and similarly for ''T''''B'' and ''T''''C''. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments ''AT''''A'', ''BT''''B'', ''CT''''C'' are called the triangle's splitters.
== Relation to other triangle centers ==
The Nagel point is the isotomic conjugate of the Gergonne point. The Nagel point, the centroid, and the incenter are collinear on a line called the ''Nagel line''. The incenter is the Nagel point of the medial triangle;〔(【引用サイトリンク】title=Why is the Incenter the Nagel Point of the Medial Triangle? )〕 equivalently, the Nagel point is the incenter of the anticomplementary triangle.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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