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 edges and three 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, any three points, when non- colli ...

''ABC'', there exist unique points ''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 pr ...

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 ...

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 t ...

References

{{reflist Triangle centers

See also

References