Trilinear coordinates
In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle.
The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z : x and vertices C and A.
In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (a', b', c'), or equivalently in ratio form, ka' : kb' : kc' for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0.
If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative.
for trilinear coordinates is often used in preference to the ordered triple notation
can be rescaled by any arbitrary value without affecting their ratio.
The trilinear coordinates of the incenter of a triangle △ABC are 1 : 1 : 1; that is, the (directed) distances from the incenter to the sidelines BC, CA, AB are proportional to the actual distances denoted by (r, r, r), where r is the inradius of △ABC.
The midpoint of, for example, side BC has trilinear coordinates in actual sideline distances
for triangle area Δ, which in arbitrarily specified relative distances simplifies to 0 : ca : ab.
The coordinates in actual sideline distances of the foot of the altitude from A to BC are
96 Trilinear coordinates enable many algebraic methods in triangle geometry.
23 From this, every straight line has a linear equation homogeneous in x, y, z.
in real coefficients is a real straight line of finite points unless l : m : n is proportional to a : b : c, the side lengths, in which case we have the locus of points at infinity.[1]: p.
40 The dual of this proposition is that the lines concur in a point (α, β, γ) if and only if D = 0.[1]: p.
28 Also, if the actual directed distances are used when evaluating the determinant of D, then the area of triangle △PUX is KD, where
The tangents of the angles between two lines with trilinear equations
49 The equation of the altitude from vertex A to side BC is[1]: p.98, #x The equation of a line with variable distances p, q, r from the vertices A, B, C whose opposite sides are a, b, c is[1]: p. 97, #viii The trilinears with the coordinate values a', b', c' being the actual perpendicular distances to the sides satisfy[1]: p. 11 for triangle sides a, b, c and area Δ.
The distance d between two points with actual-distance trilinears ai : bi : ci is given by[1]: p. 46 or in a more symmetric way The distance d from a point a' : b' : c' , in trilinear coordinates of actual distances, to a straight line
is[1]: p. 48 The equation of a conic section in the variable trilinear point x : y : z is[1]: p.118 It has no linear terms and no constant term.
The equation of a circle of radius r having center at actual-distance coordinates (a', b', c' ) is[1]: p.287 The equation in trilinear coordinates x, y, z of any circumconic of a triangle is[1]: p. 192 If the parameters l, m, n respectively equal the side lengths a, b, c (or the sines of the angles opposite them) then the equation gives the circumcircle.[1]: p.
The equation in trilinear coordinates of the circumconic with center x' : y' : z' is[1]: p. 203 Every conic section inscribed in a triangle has an equation in trilinear coordinates:[1]: p. 208 with exactly one or three of the unspecified signs being negative.
The equation of the incircle can be simplified to[1]: p. 210, p.214 while the equation for, for example, the excircle adjacent to the side segment opposite vertex A can be written as[1]: p. 215 Many cubic curves are easily represented using trilinear coordinates.
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: For any choice of trilinear coordinates x : y : z to locate a point, the actual distances of the point from the sidelines are given by a' = kx, b' = ky, c' = kz where k can be determined by the formula
in which a, b, c are the respective sidelengths BC, CA, AB, and ∆ is the area of △ABC.
Conversely, a point with barycentrics α : β : γ has trilinear 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
Similarly define the position vector of vertex A as
Then any point P associated with the reference triangle △ABC can be defined in a Cartesian system as a vector
If this point P has trilinear coordinates x : y : z then the conversion formula from the coefficients k1 and k2 in the Cartesian representation to the trilinear coordinates is, for side lengths a, b, c opposite vertices A, B, C, and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors
are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ax, by, cz as the weights.
