Mittenpunkt
Encyclopedia
In geometry
, the mittenpunkt (German
, middlespoint) of a triangle
is a triangle center
: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It is defined as the symmedian
point
of the excentral triangle
of the given triangle. It is also the centroid
of the Mandart inellipse of the given triangle.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the mittenpunkt (German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
, middlespoint) of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
is a triangle center
Triangle center
In geometry a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of...
: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It is defined as the symmedian
Symmedian
Symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle , and reflecting the line over the corresponding angle bisector...
point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
of the excentral triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
of the given triangle. It is also the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
of the Mandart inellipse of the given triangle.