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. Trilinear coordinates are an example of homogeneous coordinates. 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. It is impossible for all three trilinear coordinates to be non-positive.

Notation

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 $(a',b',c')$ 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 $x:y:z={}\!$ $2x:2y:2z$ represent the same point.

Examples

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 $△ ABC$ . Given side lengths $a, b, c$ we have:

Name; Symbol		Trilinear coordinates	Description
Vertices	$A$	$1:0:0$	Points at the corners of the triangle
	$B$	$0:1:0$
	$C$	$0:0:1$
Incenter	$I$	$1:1:1$	Intersection of the internal angle bisectors; Center of the triangle's inscribed circle
Excenters	$I A$	$-1:1:1$	Intersections of the angle bisectors (two external, one internal); Centers of the triangle's three escribed circles
	$I B$	$1:-1:1$
	$I C$	$1:1:-1$
Centroid	$G$	${\frac {1}{a}}:{\frac {1}{b}}:{\frac {1}{c}}$	Intersection of the medians; Center of mass of a uniform triangular lamina
Circumcenter	$O$	$\cos A:\cos B:\cos C$	Intersection of the perpendicular bisectors of the sides; Center of the triangle's circumscribed circle
Orthocenter	$H$	$\sec A:\sec B:\sec C$	Intersection of the altitudes
Nine-point center	$N$	${\displaystyle {\begin{aligned}&\cos(B-C):\cos(C-A)\\&\qquad$	Center of the circle passing through the midpoint of each side, the foot of each altitude, and the midpoint between the orthocenter and each vertex
Symmedian point	$K$	$a:b:c$	Intersection of the symmedians – the reflection of each median about the corresponding angle bisector

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.)

The midpoint of, for example, side $BC$ has trilinear coordinates in actual sideline distances $(0,{\tfrac {\Delta }{b}},{\tfrac {\Delta }{c}})$ 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 $(0,{\tfrac {2\Delta }{a}}\cos C,{\tfrac {2\Delta }{a}}\cos B),$ which in purely relative distances simplifies to $0 : cos C : cos B$ .^[1]^{: p. 96}

Formulas

Collinearities and concurrencies

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

{\begin{aligned}P&=p:q:r\\U&=u:v:w\\X&=x:y:z\\\end{aligned}}

are collinear if and only if the determinant

D={\begin{vmatrix}p&q&r\\u&v&w\\x&y&z\end{vmatrix}}

equals zero. Thus if $x : y : z$ is a variable point, the equation of a line through the points $P$ and $U$ is $D = 0$ .^[1]^{: p. 23} From this, every straight line has a linear equation homogeneous in $x, y, z$ . Every equation of the form $lx+my+nz=0$ 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

{\begin{aligned}p\alpha +q\beta +r\gamma &=0\\u\alpha +v\beta +w\gamma &=0\\x\alpha +y\beta +z\gamma &=0\end{aligned}}

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 $K={\tfrac {-abc}{8\Delta ^{2}}}$ (and where $Δ$ is the area of triangle $△ ABC$ , as above) if triangle $△ PUX$ has the same orientation (clockwise or counterclockwise) as $△ ABC$ , and $K={\tfrac {-abc}{8\Delta ^{2}}}$ otherwise.

Parallel lines

Two lines with trilinear equations $lx+my+nz=0$ and $l'x+m'y+n'z=0$ are parallel if and only if^[1]^{: p. 98, #xi}

{\begin{vmatrix}l&m&n\\l'&m'&n'\\a&b&c\end{vmatrix}}=0,

where $a, b, c$ are the side lengths.

Angle between two lines

The tangents of the angles between two lines with trilinear equations $lx+my+nz=0$ and $l'x+m'y+n'z=0$ are given by^[1]^: p.50

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

Perpendicular lines

Thus two lines with trilinear equations $lx+my+nz=0$ and $l'x+m'y+n'z=0$ are perpendicular if and only if

ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0.

Altitude

The equation of the altitude from vertex $A$ to side $BC$ is^[1]^{: p.98, #x}

y\cos B-z\cos C=0.

Line in terms of distances from vertices

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}

apx+bqy+crz=0.

Actual-distance trilinear coordinates

The trilinears with the coordinate values $a', b', c'$ being the actual perpendicular distances to the sides satisfy^[1]^{: p. 11}

aa'+bb'+cc'=2\Delta

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

Distance between two points

The distance $d$ between two points with actual-distance trilinears $a i : b i : c i$ is given by^[1]^{: p. 46}

d^{2}\sin ^{2}C=(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}+2(a_{1}-a_{2})(b_{1}-b_{2})\cos C

or in a more symmetric way

d^{2}={\frac {abc}{4\Delta ^{2}}}\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})\right).

Distance from a point to a line

The distance $d$ from a point $a' : b' : c'$ , in trilinear coordinates of actual distances, to a straight line $lx+my+nz=0$ is^[1]^{: p. 48}

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

Quadratic curves

The equation of a conic section in the variable trilinear point $x : y : z$ is^[1]^: 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 $r$ having center at actual-distance coordinates $(a', b', c')$ is^[1]^: p.287

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

Circumconics

The equation in trilinear coordinates $x, y, z$ of any circumconic of a triangle is^[1]^{: p. 192}

lyz+mzx+nxy=0.

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. 199}

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center $x' : y' : z'$ is^[1]^{: p. 203}

yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0.

Inconics

Every conic section inscribed in a triangle has an equation in trilinear coordinates:^[1]^{: p. 208}

l^{2}x^{2}+m^{2}y^{2}+n^{2}z^{2}\pm 2mnyz\pm 2nlzx\pm 2lmxy=0,

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}

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

while the equation for, for example, the excircle adjacent to the side segment opposite vertex $A$ can be written as^[1]^{: p. 215}

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

Cubic curves

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

{\begin{vmatrix}x&y&z\\qryz&rpzx&pqxy\\u&v&w\end{vmatrix}}=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

Between trilinear coordinates and distances from sidelines

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 $k={\tfrac {2\Delta }{ax+by+cz}}$ in which $a, b, c$ are the respective sidelengths $BC, CA, AB$ , and $∆$ is the area of $△ ABC$ .

Between barycentric and trilinear coordinates

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

Between Cartesian and 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 ${\vec {B}},$ using vertex $C$ as the origin. Similarly define the position vector of vertex $A$ as ${\vec {A}}.$ Then any point $P$ associated with the reference triangle $△ ABC$ can be defined in a Cartesian system as a vector ${\vec {P}}=k_{1}{\vec {A}}+k_{2}{\vec {B}}.$ If this point $P$ has trilinear coordinates $x : y : z$ then the conversion formula from the coefficients $k 1$ and $k 2$ in the Cartesian representation to the trilinear coordinates is, for side lengths $a, b, c$ opposite vertices $A, B, C$ ,

x:y:z={\frac {k_{1}}{a}}:{\frac {k_{2}}{b}}:{\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}={\frac {ax}{ax+by+cz}},\quad k_{2}={\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 ${\vec {A}},{\vec {B}},{\vec {C}}$ and if the point $P$ has trilinear coordinates $x : y : z$ , then the Cartesian coordinates of ${\vec {P}}$ are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates $ax, by, cz$ as the weights. Hence the conversion formula from the trilinear coordinates $x, y, z$ to the vector of Cartesian coordinates ${\vec {P}}$ of the point is given by

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

where the side lengths are

{\begin{aligned}&|{\vec {C}}-{\vec {B}}|=a,\\&|{\vec {A}}-{\vec {C}}|=b,\\&|{\vec {B}}-{\vec {A}}|=c.\end{aligned}}

References

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 William Allen Whitworth (1866) Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise, link from Cornell University Historical Math Monographs.

External links

Weisstein, Eric W. "Trilinear Coordinates". MathWorld.
Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 7000 triangle centers

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[WW-1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 William Allen Whitworth (1866) Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise, link from Cornell University Historical Math Monographs.

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