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| [[File:Lemoine_Hexagon.svg|thumb|360px|The Lemoine hexagon, shown with self-intersecting connectivity, circumscribed by the first Lemoine circle]]
| | The author's name is Andera and she believes it sounds fairly great. Some time in the past she chose to reside in Alaska and her parents live close by. Distributing manufacturing is exactly where her main earnings comes from. The favorite hobby for him and his kids is to play lacross and he would by no means give it up.<br><br>Here is my webpage ... psychic phone readings, [http://conniecolin.com/xe/community/24580 conniecolin.com], |
| In [[geometry]], the '''Lemoine hexagon''' is a [[cyclic polygon|cyclic]] [[hexagon]] with [[vertex (geometry)|vertices]] given by the six intersections of the edges of a [[triangle]] and the three lines that are parallel to the edges that pass through its [[symmedian point]]. There are two definitions of the hexagon that differ based on the order in which the vertices are connected.
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| ==Area and perimeter==
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| The Lemoine hexagon can be drawn defined in two ways, first as a simple hexagon with vertices at the intersections as defined before. The second is a self-intersecting hexagon with the lines going through the symmedian point as three of the edges and the other three edges join pairs of adjacent vertices.
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| For the simple hexagon drawn in a triangle with side lengths <math>a, b, c</math> and area <math>\Delta</math> the perimeter is given by
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| :<math>
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| p = \frac{a^3+b^3+c^3+3abc}{a^2+b^2+c^2}
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| </math>
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| and the area by | |
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| :<math>
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| a = \frac{a^4+b^4+c^4+a^2b^2+b^2c^2+c^2a^2}{\left( a^2+b^2+c^2 \right)^2} \Delta
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| </math> | |
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| For the self intersecting hexagon the perimeter is given by
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| :<math>
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| p = \frac{\left( a+b+c\right) \left(ab+bc+ca\right)}{a^2+b^2+c^2}
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| </math>
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| and the area by
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| :<math>
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| a = \frac{a^2b^2+b^2c^2+c^2a^2}{\left(a^2+b^2+c^2\right)^2}\Delta
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| </math>
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| ==Circumcircle==
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| In geometry, [[five points determine a conic]], so arbitrary sets of six points to not generally lie on a conic section, let alone a circle. Nevertheless, the Lemoine hexagon (with either order of connection) is a [[cyclic polygon]], meaning that its vertices all lie on a common circle. The circumcircle of the Lemoine hexagon is known as the '''first Lemoine circle'''.
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| ==References==
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| *{{citation|last=Casey|first=John|authorlink=John Casey (mathematician)|title=A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples|edition=5th|location=Dublin|publisher=Hodges, Figgis, & Co.|year=1888|chapter=Lemoine's, Tucker's, and Taylor's Circles|pages=179ff|url=http://books.google.com/books?id=i87Ikm2u_6sC&pg=PA179}}.
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| *{{citation|first=É.|last=Lemoine|authorlink=Émile Lemoine|contribution=Sur quelques propriétés d’un point remarquable d’un triangle|title=Association francaise pour l’avancement des sciences, Congrès (002; 1873; Lyon)|year=1874|pages=90–95|language=French|url=http://gallica.bnf.fr/ark:/12148/bpt6k201149r/f128.image}}.
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| *{{citation|first=J. S.|last=Mackay|title=Symmedians of a triangle and their concomitant circles|journal=Proceedings of the Edinburgh Mathematical Society|volume=14|year=1895|pages=37–103|doi=10.1017/S0013091500031758}}.
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| == External links ==
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| *{{mathworld|id=LemoineHexagon|title=Lemoine Hexagon}}
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| [[Category:Polygons]]
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The author's name is Andera and she believes it sounds fairly great. Some time in the past she chose to reside in Alaska and her parents live close by. Distributing manufacturing is exactly where her main earnings comes from. The favorite hobby for him and his kids is to play lacross and he would by no means give it up.
Here is my webpage ... psychic phone readings, conniecolin.com,