Trilinear coordinates

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates. They are often called simply "trilinears".

Examples

The incenter has trilinears 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines "BC", "CA", "AB" of a triangle "ABC" are proportional to the actual distances, which are the ordered triple ("r", "r", "r"), where "r" is the inradius of triangle "ABC". Note that the notation "x":"y":"z" using colons distinguishes trilinears from actual directed distances, ("kx", "ky", "kz"), which is the usual notation for an ordered triple, and which may be obtained from "x" : "y" : "z" using the number

: k = frac{2sigma}{ax + by + cz}

where "a", "b", "c" are the respective sidelengths "BC", "CA", "AB", and σ = area of "ABC". ("Comma notation" for trilinears should be avoided, because the notation ("x", "y", "z"), which means an ordered triple, does not allow, for example, ("x", "y", "z") = (2"x", 2"y", 2"z"), whereas the "colon notation" does allow "x" : "y" : "z" = 2"x" : 2"y" : 2"z".)

Let "A", "B", and "C" be either the vertices of the triangle, or the corresponding angles at those vertices. Trilinears for several well known points are:

:* "A" = 1 : 0 : 0:* "B" = 0 : 1 : 0:* "C" = 0 : 0 : 1:* incenter = 1 : 1 : 1:* centroid = "bc" : "ca" : "ab" = 1/"a" : 1/"b" : 1/"c" = csc "A" : csc "B" : csc "C".:* circumcenter = cos "A" : cos "B" : cos "C".:* orthocenter = sec "A" : sec "B" : sec "C".:* nine-point center = cos("B" − "C") : cos("C" − "A") : cos("A" − "B").:* symmedian point = "a" : "b" : "c" = sin "A" : sin "B" : sin "C".:* "A"-excenter = −1 : 1 : 1:* "B"-excenter = 1 : −1 : 1:* "C"-excenter = 1 : 1 : −1

Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas of the triangles "BGC", "CGA", "AGB", where "G" = centroid.)

Formulas

Trilinears enable many algebraic methods in triangle geometry. For example, three points

:"P = p" : "q" : "r":"U = u" : "v" : "w":"X = x" : "y" : "z"

are collinear if and only if the determinant

: D = egin{bmatrix}p&q&r\u&v&w\x&y&zend{bmatrix}.

equals zero. The dual of this proposition is that the lines

:"pα + qβ + rγ = 0":"uα + vβ + wγ = 0",:"xα + yβ + zγ = 0"

concur in a point if and only if "D = 0."

Also, (area of "(PUX)) = KD", where "K = abc/8σ""2" if triangle PUX has the same orientation as triangle ABC, and "K = - abc/8σ""2" otherwise.

Many cubic curves are easily represented using trilinears. For example, the pivotal self-isoconjugate cubic "Z(U,P)", as the locus of a point "X" such that the "P"-isoconjugate of "X" is on the line "UX" is given by the determinant equation

: egin{bmatrix}x&y&z\qryz&rpzx&pqxy\u&v&wend{bmatrix} = 0.

Among named cubics "Z(U,P)" are the following:

: Thomson cubic: "Z(X(2),X(1))", where "X(2) = "centroid, "X(1) = "incenter: Feuerbach cubic: "Z(X(5),X(1))", where "X(5) = "Feuerbach point: Darboux cubic: "Z(X(20),X(1))", where "X(20) = "De Longchamps point : Neuberg cubic: "Z(X(30),X(1))", where "X(30) = "Euler infinity point

Conversions

A point with trilinears "α" : "β" : "γ" has barycentric coordinates "aα" : "bβ" : "cγ" where "a", "b", "c" are the sidelengths of the triangle. Conversely, a point with barycentrics "α" : "β" : "γ" has trilinears "α/a" : "β/b" : "γ/c".

There are formulas for converting between trilinear coordinates and 2D Cartesian coordinates. Given a reference triangle ABC express the position of the vertex B in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector "a" using vertex C as the origin. Similarly define the position vector of vertex A as "b". Then any point P associated with the reference triangle ABC can be defined in a 2D Cartesian system as a vector "p" = α"a" + β"b". If this point P has trilinear coordinates x : y : z then the conversion formulas are as follows:

: x:y:z = frac{eta}{a} : frac{alpha}{b} : frac{1 - alpha - eta}{c}

alternatively

: alpha = frac{by}{ax + by + cz} mbox{ and } eta = frac{ax}{ax + by + cz}.

External links

* [http://mathworld.wolfram.com/TrilinearCoordinates.html Trilinear Coordinates] on Mathworld.
* [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers - ETC] by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 3200 triangle centers


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