In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, given a
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
''ABC'', there exist unique
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
''A´'', ''B´'', and ''C´'' on the sides ''BC'', ''CA'', ''AB'' respectively, such that:
:* ''A´'', ''B´'', and ''C´'' partition the
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
of the triangle into three equal-length pieces. That is,
:::.
:* The three lines ''AA´'', ''BB´'', and ''CC´'' meet in a point, the trisected perimeter point.
This is point ''X''
369 in Clark Kimberling's ''Encyclopedia of Triangle Centers''.
[Kimberling, C. ''Encyclopedia of Triangle Centers'']
X(369) = 1st TRISECTED PERIMETER POINT
Uniqueness and a formula for the
trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
of ''X''
369 were shown by Peter Yff late in the twentieth century. The formula involves the unique real root of a
cubic equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
.
[
]
See also
* Bisected perimeter point
References
{{reflist
Triangle centers