Splitter (geometry)

In Euclidean 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] They are not to be confused with cleavers, which also bisect the perimeter but instead emanate from the midpoint of one of the triangle's sides.

  Arbitrary triangle △ABC
  Excircles, tangent to the sides of △ABC at TA, TB, TC
  Extouch triangle △TATBTC
  Splitters of the perimeter ATA, BTB, CTC; intersect at the Nagel point N

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. Honsberger, Ross (1995), "Chapter 1: Cleavers and Splitters", Episodes in Nineteenth and Twentieth Century Euclidean Geometry, New Mathematical Library, vol. 37, Washington, DC: Mathematical Association of America, pp. 1–14, ISBN 0-88385-639-5, MR 1316889
  2. 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
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