- Trilinear coordinates
In
geometry , the trilinear coordinates of a point relative to a giventriangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example ofhomogeneous 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 : −1Note 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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