In geometry

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 Nagel point is a point associated with any 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 ....

. Given a triangle ABC, let T_A, T_B, and T_C be the extouch points

Extouch triangle

The extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. The vertices of the extouch triangle are given in trilinear coordinates by:...

 in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively. The lines AT_A, BT_B, CT_C concur

Concurrent lines

In geometry, two or more lines are said to be concurrent if they intersect at a single point.In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:...

 in the Nagel point N of triangle ABC. The Nagel point is named after Christian Heinrich von Nagel

Christian Heinrich von Nagel

Christian Heinrich von Nagel was a German geometer.After the visit of the gymnasium in 1817, Nagel went to Evangelical Seminaries of Maulbronn and Blaubeuren. From 1821 to 1825, he took a four-year course of theology at the Tübinger Stift. After his graduation, he early became interested in...

, a nineteenth century German mathematician, who wrote about it in 1836.

Another construction of the point T_A is to start at A and trace around triangle ABC half its perimeter

Semiperimeter

In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name...

, and similarly for T_B and T_C. Because of this construction, the Nagel point is sometimes also called the bisected perimeter point.

Relation to other triangle centers

The Nagel point is the isotomic conjugate

Isotomic conjugate

In geometry, the isotomic conjugate of a point P not on a sideline of triangle ABC is constructed as follows: Let A, B, C be the points in which the lines AP, BP, CP meet the lines BC, CA, AB, respectively. Reflect ABC in the midpoints of sides BC, CA, AB to obtain points A", B", C", respectively...

 of the Gergonne point. The Nagel point, the incenter, and 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...

 are collinear. The incenter is the Nagel point of the medial triangle

Medial triangle

The medial triangle of a triangle ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC and BC....

 (Anonymous 1896); equivalently, the Nagel point is the incenter of the anticomplementary triangle.

Trilinear coordinates

Trilinear coordinates

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...

 of the Nagel point were given by Gallatly (1913) as


or, equivalently, in terms of the side lengths a = |BC|, b = |CA|, and c = |AB|,

External links

• Nagel Point from Cut-the-knot
  Cut-the-knot
  Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
  
  
• Nagel Point, Clark Kimberling
• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.