Nagel point

The lines ATA, BTB, CTC concur in the Nagel point N of triangle △ABC.

Another construction of the point TA is to start at A and trace around triangle △ABC half its perimeter, and similarly for TB and TC.

Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments ATA, BTB, CTC are called the triangle's splitters.

There exists an easy construction of the Nagel point.

Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge.

The incenter is the Nagel point of the medial triangle;[2][3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.

The trilinear coordinates of the Nagel point are[4] as or, equivalently, in terms of the side lengths

Early contributions to the study of this point were also made by August Leopold Crelle and Carl Gustav Jacob Jacobi.

Arbitrary triangle △ ABC
Excircles , tangent to the sides of △ ABC at T A , T B , T C
Extouch triangle △ T A T B T C
Splitters of the perimeter AT A , BT B , CT C ; intersect at the Nagel point N
Easy construction of the Nagel point