Splitter (geometry)

The Nagel point (blue, N) of a triangle (black). The red triangle is the extouch triangle, and the orange circles are the excircles. The splitters of the perimeter are ATA, BTB, and CTC.

In plane geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle.[1][2]

Properties

The opposite endpoint of a splitter to the chosen triangle vertex lies at the point on the triangle's side where one of the excircles of the triangle is tangent to that side.[1][2] This point is also called a splitting point of the triangle.[2] It is additionally a vertex of the extouch triangle and one of the points where the Mandart inellipse is tangent to the triangle side.[3]

The three splitters concur at the Nagel point of the triangle,[1] which is also called its splitting center.[2]

Generalization

Some authors have used the term "splitter" in a more general sense, for any line segment that bisects the perimeter of the triangle. Other line segments of this type include the cleavers, which are perimeter-bisecting segments that pass through the midpoint of a triangle side, and the equalizers, segments that bisect both the area and perimeter of a triangle.[4]

References

  1. 1 2 3 Honsberger, Ross (1995), "Chapter 1: Cleavers and Splitters", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, pp. 1–14.
  2. 1 2 3 4 Avishalom, Dov (1963), "The perimetric bisection of triangles", Mathematics Magazine, 36 (1): 60–62, JSTOR 2688140, MR 1571272.
  3. ↑ Juhász, Imre (2012), "Control point based representation of inellipses of triangles" (PDF), Annales Mathematicae et Informaticae, 40: 37–46, MR 3005114.
  4. ↑ Kodokostas, Dimitrios (2010), "Triangle equalizers", Mathematics Magazine, 83 (2): 141–146, doi:10.4169/002557010X482916.

External links

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