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".
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[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
- k = 2σ/(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) = (2x, 2y, 2z), whereas the "colon notation" does allow x : y : z = 2x : 2y : 2z.)
Trilinears for several well known points are as follows:
-
- A = 1 : 0 : 0
- B = 0 : 1 : 0
- C = 0 : 0 : 1
- centroid = bc : ca : ab = 1/a : 1/b : 1/c = csc A : csc B : csc B.
- 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).
- 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 conics 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.
[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
