“Lemoine hexagon”的意思、由来-开放百科全书

词条

Lemoine hexagon

释义

Area and perimeter
Circumcircle
References
External links

In geometry, the Lemoine hexagon is a cyclic hexagon with 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.

Area and perimeter

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.

For the simple hexagon drawn in a triangle with side lengths and area the perimeter is given by

and the area by

For the self intersecting hexagon the perimeter is given by

and the area by

Circumcircle

In geometry, five points determine a conic, so arbitrary sets of six points do 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.

References

{{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=https://books.google.com/books?id=i87Ikm2u_6sC&pg=PA179}}.
{{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}}.
{{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}}.

External links

{{mathworld|id=LemoineHexagon|title=Lemoine Hexagon}}

1 : Polygons

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。