Mittenpunkt

The mittenpunkt M of the black triangle, at the center of its Mandart inellipse (red). The blue lines through the middenpunkt pass through the triangle's excenters and corresponding edge midpoints.

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 was identified in 1836 by Christian Heinrich von Nagel as the symmedian point of the excentral triangle of the given triangle.^[1][2] 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 Gergonne point.^[3] The mittenpunkt is also the centroid of the Mandart inellipse of the given triangle.^[4]

References

1. Kimberling, Clark (1994), "Central Points and Central Lines in the Plane of a Triangle", Mathematics Magazine 67 (3): 163–187, doi:10.2307/2690608, JSTOR 2690608?origin=pubexport, MR 1573021.
2. v. Nagel, C. H. (1836), Untersuchungen über die wichtigsten zum Dreiecke gehörenden Kreise, Leipzig.
3. Eddy, Roland H. (1989), "A Desarguesian dual for Nagel's middlespoint", Elemente der Mathematik 44 (3): 79–80, MR 999636.
4. Gibert, Bernard (2004), "Generalized Mandart conics" (PDF), Forum Geometricorum 4: 177–198, MR 2130231.

External links

• Weisstein, Eric W., "Mittenpunkt", MathWorld.