In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, a splitter is a
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
through one of the
vertices of a
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
(that is, a
cevian) that
bisects the perimeter of the triangle.
They are not to be confused with
cleavers, which also bisect the perimeter but instead emanate from the
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dim ...
of one of the triangle's sides.
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.
This point is also called a splitting point of the triangle.
It is additionally a vertex of the
extouch triangle and one of the points where the
Mandart inellipse is tangent to the triangle side.
The three splitters
concur at the
Nagel point of the triangle,
which is also called its splitting center.
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.
References
External links
*{{mathworld, id=Splitter, title=Splitter, mode=cs2
Straight lines defined for a triangle