Trilinear coordinates

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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".

Contents 1 Examples 2 Formulas 3 Conversions 4 External links

[edit] 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

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) = (2x, 2y, 2z), whereas the "colon notation" does allow x : y : z = 2x : 2y : 2z.)

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.)

[edit] 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

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/8σ² if triangle PUX has the same orientation as triangle ABC, and K = - abc/8σ² 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

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

[edit] 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 = αa + βb. If this point P has trilinear coordinates x : y : z then the conversion formulas are as follows:

alternatively

[edit] External links

• Trilinear Coordinates on Mathworld.
• Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 3200 triangle centers