Trilinear coordinates

The ratio notation $x\; :\; y\; :\; z$ for trilinear coordinates is often used in preference to the ordered triple notation $(x,\; y,\; z),$ with the latter reserved for triples of directed distances $(a\text{'},\; b\text{'},\; c\text{'})$ relative to a specific triangle. The trilinear coordinates $x\; :\; y\; :\; z,$ can be rescaled by any arbitrary value without affecting their ratio. The bracketed, comma-separated triple notation $(x,\; y,\; z)$ can cause confusion because conventionally this represents a different triple than e.g. $(2x,\; 2y,\; 2z),$ but these equivalent ratios $x\; :\; y\; :\; z\; =\; \{\}\backslash !$$2x\; :\; 2y\; :\; 2z$ represent the same point.

The trilinear coordinates of the incenter of a triangle {{math|△ABC}} are {{math|1 : 1 : 1}}; that is, the (directed) distances from the incenter to the sidelines {{mvar|BC, CA, AB}} are proportional to the actual distances denoted by {{math|(r, r, r)}}, where {{mvar|r}} is the inradius of {{math|△ABC}}. Given side lengths {{mvar|a, b, c}} we have:

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

The midpoint of, for example, side {{mvar|BC}} has trilinear coordinates in actual sideline distances $(0\; ,\; \backslash tfrac\{\backslash Delta\}\{b\}\; ,\; \backslash tfrac\{\backslash Delta\}\{c\})$ for triangle area {{math|Δ}}, which in arbitrarily specified relative distances simplifies to {{math|0 : ca : ab}}. The coordinates in actual sideline distances of the foot of the altitude from {{mvar|A}} to {{mvar|BC}} are $(0,\; \backslash tfrac\{2\backslash Delta\}\{a\}\backslash cos\; C,\; \backslash tfrac\{2\backslash Delta\}\{a\}\backslash cos\; B),$ which in purely relative distances simplifies to {{math|0 : cos C : cos B}}.{{rp|p. 96}}

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

:$\backslash begin\{align\}$

P &= p:q:r \\

U &= u:v:w \\

X &= x:y:z \\

\end{align}

are collinear if and only if the determinant

:$D\; =\; \backslash begin\{vmatrix\}$

p & q & r \\

u & v & w \\

x & y & z

\end{vmatrix}

equals zero. Thus if {{math|x : y : z}} is a variable point, the equation of a line through the points {{mvar|P}} and {{mvar|U}} is {{math|1=D = 0}}.William Allen Whitworth (1866) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01190002 Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise], link from Cornell University Historical Math Monographs.{{rp|p. 23}} From this, every straight line has a linear equation homogeneous in {{mvar|x, y, z}}. Every equation of the form $lx+my+nz=0$ in real coefficients is a real straight line of finite points unless {{math|l : m : n}} is proportional to {{math|a : b : c}}, the side lengths, in which case we have the locus of points at infinity.{{rp|p. 40}}

The dual of this proposition is that the lines

:$\backslash begin\{align\}$

p\alpha + q\beta + r\gamma &= 0 \\

u\alpha + v\beta + w\gamma &= 0 \\

x\alpha + y\beta + z\gamma &= 0

\end{align}

concur in a point {{math|(α, β, γ)}} if and only if {{math|1=D = 0}}.{{rp|p. 28}}

Also, if the actual directed distances are used when evaluating the determinant of {{mvar|D}}, then the area of triangle {{math|△PUX}} is {{mvar|KD}}, where $K\; =\; \backslash tfrac\{-abc\}\{8\backslash Delta^2\}$ (and where {{math|Δ}} is the area of triangle {{math|△ABC}}, as above) if triangle {{math|△PUX}} has the same orientation (clockwise or counterclockwise) as {{math|△ABC}}, and $K\; =\; \backslash tfrac\{-abc\}\{8\backslash Delta^2\}$ otherwise.

Two lines with trilinear equations $lx+my+nz=0$ and $l\text{'}x+m\text{'}y+n\text{'}z=0$ are parallel if and only if{{rp|p. 98,#xi}}

:$\backslash begin\{vmatrix\}$

l & m & n \\

l' & m' & n' \\

a & b & c

\end{vmatrix}=0,

where {{mvar|a, b, c}} are the side lengths.

The tangents of the angles between two lines with trilinear equations $lx+my+nz=0$ and $l\text{'}x+m\text{'}y+n\text{'}z=0$ are given by{{rp|at=Art. 48}}

:$\backslash pm\; \backslash frac\{(mn\text{'}-m\text{'}n)\backslash sin\; A\; +\; (nl\text{'}-n\text{'}l)\backslash sin\; B\; +\; (lm\text{'}-l\text{'}m)\backslash sin\; C\}\{ll\text{'}\; +\; mm\text{'}\; +\; nn\text{'}\; -\; (mn\text{'}+m\text{'}n)\backslash cos\; A\; -(nl\text{'}+n\text{'}l)\backslash cos\; B\; -\; (lm\text{'}+l\text{'}m)\backslash cos\; C\}.$

Thus they are perpendicular if{{rp|at=Art. 49}}

:$ll\text{'}+mm\text{'}+nn\text{'}-(mn\text{'}+m\text{'}n)\backslash cos\; A-(nl\text{'}+n\text{'}l)\backslash cos\; B-(lm\text{'}+l\text{'}m)\backslash cos\; C=0.$

The equation of the altitude from vertex {{mvar|A}} to side {{mvar|BC}} is{{rp|p.98,#x}}

:$y\backslash cos\; B-z\backslash cos\; C=0.$

The equation of a line with variable distances {{mvar|p, q, r}} from the vertices {{mvar|A, B, C}} whose opposite sides are {{mvar|a, b, c}} is{{rp|p. 97,#viii}}

:$apx+bqy+crz=0.$

The trilinears with the coordinate values {{mvar|a', b', c'}} being the actual perpendicular distances to the sides satisfy{{rp|p. 11}}

:$aa\text{'}\; +bb\text{'}\; +\; cc\text{'}\; =2\backslash Delta$

for triangle sides {{mvar|a, b, c}} and area {{math|Δ}}. This can be seen in the figure at the top of this article, with interior point {{mvar|P}} partitioning triangle {{math|△ABC}} into three triangles {{math|△PBC, △PCA, △PAB}} with respective areas $\backslash tfrac\{1\}\{2\}aa\text{'}\; ,\; \backslash tfrac\{1\}\{2\}bb\text{'},\; \backslash tfrac\{1\}\{2\}cc\text{'}.$

The distance {{mvar|d}} between two points with actual-distance trilinears {{math|a{{sub|i}} : b{{sub|i}} : c{{sub|i}}}} is given by{{rp|p. 46}}

:$d^2\backslash sin\; ^2\; C=(a\_1-a\_2)^2+(b\_1-b\_2)^2+2(a\_1-a\_2)(b\_1-b\_2)\backslash cos\; C$

or in a more symmetric way

:$d^2\; =\; \backslash frac\{a\; b\; c\}\{4\backslash Delta^2\}\backslash left(a(b\_1-b\_2)(c\_2-c\_1)+b(c\_1-c\_2)(a\_2-a\_1)+c(a\_1-a\_2)(b\_2-b\_1)\backslash right).$

The distance {{mvar|d}} from a point {{math|a' : b' : c' }}, in trilinear coordinates of actual distances, to a straight line $lx+my+nz=0$ is{{rp|p. 48}}

:$d=\backslash frac\{la\text{'}+mb\text{'}+nc\text{'}\}\{\backslash sqrt\{l^2+m^2+n^2-2mn\backslash cos\; A\; -2nl\backslash cos\; B\; -2lm\backslash cos\; C\}\}.$

The equation of a conic section in the variable trilinear point {{math|x : y : z}} is{{rp|p.118}}

:$rx^2+sy^2+tz^2+2uyz+2vzx+2wxy=0.$

It has no linear terms and no constant term.

The equation of a circle of radius {{mvar|r}} having center at actual-distance coordinates {{math|(a', b', c' )}} is{{rp|p.287}}

:$(x-a\text{'})^2\backslash sin\; 2A+(y-b\text{'})^2\backslash sin\; 2B+(z-c\text{'})^2\backslash sin\; 2C=2r^2\backslash sin\; A\backslash sin\; B\backslash sin\; C.$

The equation in trilinear coordinates {{mvar|x, y, z}} of any circumconic of a triangle is{{rp|p. 192}}

:$lyz+mzx+nxy=0.$

If the parameters {{mvar|l, m, n}} respectively equal the side lengths {{mvar|a, b, c}} (or the sines of the angles opposite them) then the equation gives the circumcircle.{{rp|p. 199}}

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center {{math|x' : y' : z' }} is{{rp|p. 203}}

:$yz(x\text{'}-y\text{'}-z\text{'})+zx(y\text{'}-z\text{'}-x\text{'})+xy(z\text{'}-x\text{'}-y\text{'})=0.$

Every conic section inscribed in a triangle has an equation in trilinear coordinates:{{rp|p. 208}}

:$l^2x^2+m^2y^2+n^2z^2\; \backslash pm\; 2mnyz\; \backslash pm\; 2nlzx\backslash pm\; 2lmxy\; =0,$

with exactly one or three of the unspecified signs being negative.

The equation of the incircle can be simplified to{{rp|p. 210, p.214}}

:$\backslash pm\; \backslash sqrt\{x\}\backslash cos\; \backslash frac\{A\}\{2\}\backslash pm\; \backslash sqrt\{y\}\backslash cos\; \backslash frac\{B\}\{2\}\backslash pm\backslash sqrt\{z\}\backslash cos\; \backslash frac\{C\}\{2\}=0,$

while the equation for, for example, the excircle adjacent to the side segment opposite vertex {{mvar|A}} can be written as{{rp|p. 215}}

:$\backslash pm\; \backslash sqrt\{-x\}\backslash cos\; \backslash frac\{A\}\{2\}\backslash pm\; \backslash sqrt\{y\}\backslash cos\; \backslash frac\{B\}\{2\}\backslash pm\backslash sqrt\{z\}\backslash cos\; \backslash frac\{C\}\{2\}=0.$

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

:

qryz&rpzx&pqxy\\

u&v&w\end{vmatrix} = 0.

Among named cubics {{math|Z(U, P)}} are the following:

: Thomson cubic: {{tmath|Z(X(2),X(1))}}, where {{tmath|X(2)}} is centroid and {{tmath|X(1)}} is incenter

: Feuerbach cubic: {{tmath|Z(X(5),X(1))}}, where {{tmath|X(5)}} is Feuerbach point

: Darboux cubic: {{tmath|Z(X(20),X(1))}}, where {{tmath|X(20)}} is De Longchamps point

: Neuberg cubic: {{tmath|Z(X(30),X(1))}}, where {{tmath|X(30)}} is Euler infinity point.

For any choice of trilinear coordinates {{math|x : y : z}} to locate a point, the actual distances of the point from the sidelines are given by {{math|1=a' = kx, b' = ky, c' = kz}} where {{mvar|k}} can be determined by the formula $k\; =\; \backslash tfrac\{2\backslash Delta\}\{ax\; +\; by\; +\; cz\}$ in which {{mvar|a, b, c}} are the respective sidelengths {{mvar|BC, CA, AB}}, and {{math|∆}} is the area of {{math|△ABC}}.

A point with trilinear coordinates {{math|x : y : z}} has barycentric coordinates {{math|ax : by : cz}} where {{mvar|a, b, c}} are the sidelengths of the triangle. Conversely, a point with barycentrics {{math|α : β : γ}} has trilinear coordinates $\backslash tfrac\{\backslash alpha\}\{a\}\; :\; \backslash tfrac\{\backslash beta\}\{b\}\; :\; \backslash tfrac\{\backslash gamma\}\{c\}.$

Given a reference triangle {{math|△ABC}}, express the position of the vertex {{mvar|B}} in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector {{tmath|\vec B,}} using vertex {{mvar|C}} as the origin. Similarly define the position vector of vertex {{mvar|A}} as {{tmath|\vec A.}} Then any point {{mvar|P}} associated with the reference triangle {{math|△ABC}} can be defined in a Cartesian system as a vector $\backslash vec\; P\; =\; k\_1\backslash vec\; A\; +\; k\_2\backslash vec\; B.$ If this point {{mvar|P}} has trilinear coordinates {{math|x : y : z}} then the conversion formula from the coefficients {{math|k₁}} and {{math|k₂}} in the Cartesian representation to the trilinear coordinates is, for side lengths {{mvar|a, b, c}} opposite vertices {{mvar|A, B, C}},

: $x:y:z\; =\; \backslash frac\{k\_1\}\{a\}\; :\; \backslash frac\{k\_2\}\{b\}\; :\; \backslash frac\{1\; -\; k\_1\; -\; k\_2\}\{c\},$

and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is

: $k\_1\; =\; \backslash frac\{ax\}\{ax\; +\; by\; +\; cz\},\; \backslash quad\; k\_2\; =\; \backslash frac\{by\}\{ax\; +\; by\; +\; cz\}.$

More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors {{tmath|\vec A, \vec B, \vec C}} and if the point {{mvar|P}} has trilinear coordinates {{math|x : y : z}}, then the Cartesian coordinates of {{tmath|\vec P}} are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates {{mvar|ax, by, cz}} as the weights. Hence the conversion formula from the trilinear coordinates {{mvar|x, y, z}} to the vector of Cartesian coordinates {{tmath|\vec P}} of the point is given by

: $\backslash vec\{P\}=\backslash frac\{ax\}\{ax+by+cz\}\backslash vec\{A\}+\backslash frac\{by\}\{ax+by+cz\}\backslash vec\{B\}+\backslash frac\{cz\}\{ax+by+cz\}\backslash vec\{C\},$

where the side lengths are

:$\backslash begin\{align\}$

& |\vec C - \vec B| = a, \\

& |\vec A - \vec C| = b, \\

& |\vec B - \vec A| = c.

\end{align}

• Morley's triangles, giving examples of numerous points expressed in trilinear coordinates
• Ternary plot
• Viviani's theorem

{{Reflist}}

• {{MathWorld|title=Trilinear Coordinates|urlname=TrilinearCoordinates}}
• [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers - ETC] by Clark Kimberling; has trilinear coordinates (and barycentric) for 64000 triangle centers.

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Category:Linear algebra

Category:Affine geometry

Category:Triangle geometry

Category:Coordinate systems