geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Lemoine hexagon is a

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

with vertices given by the six intersections of the edges of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

and the three lines that are parallel to the edges that pass through its

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

. 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

a, b, c

and area

\Delta

the perimeter is given by :

p = \frac

and the area by :

K = \frac \Delta

For the self intersecting hexagon the perimeter is given by :

p = \frac

and the area by :

K = \frac\Delta

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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...

, meaning that its vertices all lie on a common circle. The circumcircle of the Lemoine hexagon is known as the first Lemoine circle.

References

*. *. *.

External links

*{{mathworld, id=LemoineHexagon, title=Lemoine Hexagon Types of polygons