geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the Lemoine hexagon is a cyclic

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 is de ...

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

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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

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 In Euclidean geometry, Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) Point (geometry), points determine a line (geometry), line (a degree-1 plane curve). There are additional subt ...

, 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, a set (mathematics), set of point (geometry), points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertex (geometry), vertices are concyclic is called a cyclic polygon, and the circle is cal ...

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

References

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External links

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