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[[File:Mandart inellipse - resized.svg|thumb|300px|Mandart inellipse (red), touching the triangle (black) in contact points of excircles (gray). In green, lines identifying Nagel point (N) and, in blue, lines identifying mittenpunkt (M).]]
In [[geometry]], the '''Mandart inellipse''' of a [[triangle]] is an [[ellipse]] inscribed within the triangle, tangent to its sides at the contact points of its [[excircle]]s (which are also the vertices of the [[extouch triangle]]).<ref>{{citation
| last = Juhász | first = Imre
| journal = Annales Mathematicae et Informaticae
| mr = 3005114
| pages = 37–46
| title = Control point based representation of inellipses of triangles
| url = http://ami.ektf.hu/uploads/papers/finalpdf/AMI_40_from37to46.pdf
| volume = 40
| year = 2012}}.</ref> The Mandard inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century.<ref name="g04">{{citation|last=Gibert|first=Bernard|title=Generalized Mandart conics|journal=Forum Geometricorum|url=http://forumgeom.fau.edu/FG2004volume4/FG200421.pdf|year=2004|volume=4|pages=177–198}}.</ref><ref>{{citation|last=Mandart|first=H.|title=Sur l’hyperbole de Feuerbach|journal=[[Mathesis (journal)|Mathesis]]|year=1893|pages=81–89}};
{{citation|last=Mandart|first=H.|title=Sur une ellipse associée au triangle|journal=[[Mathesis (journal)|Mathesis]]|year=1894|pages=241–245|url=http://books.google.com/books?id=kqAKAAAAYAAJ&pg=PA241}}. As cited by {{harvtxt|Gibert|2004}}.</ref>
 
==Parameters==
As an [[inconic]], the Mandart inellipse is described by the [[parameters]]
:<math>x:y:z=\frac{a}{b+c-a}:\frac{b}{a+c-b}:\frac{c}{a+b-c}</math>
where ''a'', ''b'', and ''c'' are sides of the given triangle.
 
==Related points==
The center of the Mandart inellipse is the [[mittenpunkt]] of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the [[Nagel point]] of the triangle.<ref name="g04"/>
 
==Notes==
{{Reflist}}
 
==External links==
*{{mathworld|urlname=MandartInellipse|title=Mandart Inellipse}}
 
[[Category:Triangle geometry]]

Latest revision as of 00:40, 10 September 2013

Mandart inellipse (red), touching the triangle (black) in contact points of excircles (gray). In green, lines identifying Nagel point (N) and, in blue, lines identifying mittenpunkt (M).

In geometry, the Mandart inellipse of a triangle is an ellipse inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle).[1] The Mandard inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century.[2][3]

Parameters

As an inconic, the Mandart inellipse is described by the parameters

x:y:z=ab+ca:ba+cb:ca+bc

where a, b, and c are sides of the given triangle.

Related points

The center of the Mandart inellipse is the mittenpunkt of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the Nagel point of the triangle.[2]

Notes

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