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 (from

German German(s) may refer to: * Germany, the country of the Germans and German things **Germania (Roman era) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizenship in Germany, see also Ge ...

: ''middle point'') 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 ...

is a

triangle center In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...

: a point defined from the triangle that is invariant under

Euclidean transformation In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations ...

s of the triangle. It was identified in 1836 by Christian Heinrich von Nagel as the symmedian point of the excentral triangle of the given triangle..

Coordinates

The mittenpunkt has

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

(b+c-a): (c+a-b ):(a+b-c)

where , , and are the side lengths of the given triangle. Expressed instead in terms of the angles , , and , the trilinears arehttp://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers :

\cot \frac : \cot \frac : \cot \frac=(\csc A+\cot A):(\csc B+\cot B):(\csc C+\cot C).

The barycentric coordinates are :

a(b+c-a):b(c+a-b):c(a+b-c) = (1+\cos A):(1+\cos B):(1+\cos C).

Collinearities

The mittenpunkt is at the intersection of the line connecting the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

and the

Gergonne point In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

, the line connecting the

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...

and the

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 ...

and the line connecting the

orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

with the

Spieker center In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in t ...

, thus establishing three collinearities involving the mittenpunkt.

Related figures

The three lines connecting the excenters of the given triangle to the corresponding edge midpoints all meet at the mittenpunkt; thus, it is the center of perspective of the excentral triangle and the median triangle, with the corresponding axis of perspective being the trilinear polar of the

. The mittenpunkt is also the

of the Mandart inellipse of the given triangle, the ellipse tangent to the triangle at its extouch points..

Notes

The Mittenpunkt also serves as the

of the

Medial triangle In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is no ...

References

External links

*{{mathworld, urlname=Mittenpunkt, title=Mittenpunkt Triangle centers