Mittenpunkt

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Arbitrary triangle

  Mandart inellipse (centered at the mittenpunkt $M$ )

  Lines from the triangle's excenters to each corresponding edge midpoint (concurrent at the mittenpunkt $M$ )

  Splitters of the triangle (concurrent at the Nagel point $N$ )

In geometry, the mittenpunkt (from German: middle point) of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations 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.^[1]^[2]

The mittenpunkt has trilinear coordinates^[1]

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

where $a$ , $b$ , and $c$ are the side lengths of the given triangle. Expressed instead in terms of the angles $A$ , $B$ , and $C$ , the trilinears are^[3]

\cot {\frac {A}{2}}:\cot {\frac {B}{2}}:\cot {\frac {C}{2}}=(\csc A+\cot A):(\csc B+\cot B):(\csc C+\cot C).

The barycentric coordinates are^[3]

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 and the Gergonne point, the line connecting the incenter and the symmedian point and the line connecting the orthocenter with the Spieker center, thus establishing three collinearities involving the mittenpunkt.^[4]

Related figures

References

External links

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