Trilinear coordinates

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In geometry, the trilinear coordinates Template:Math 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 Template:Math is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices Template:Mvar and Template:Mvar respectively; the ratio Template:Math is the ratio of the perpendicular distances from the point to the sidelines opposite vertices Template:Mvar and Template:Mvar respectively; and likewise for Template:Math and vertices Template:Mvar and Template:Mvar.

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (Template:Mvar, Template:Mvar, Template:Mvar), or equivalently in ratio form, Template:Math for any positive constant Template:Mvar. 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 $(a^{\prime },b^{\prime },c^{\prime })$ 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 Template:Math are Template:Math; that is, the (directed) distances from the incenter to the sidelines Template:Mvar are proportional to the actual distances denoted by Template:Math, where Template:Mvar is the inradius of Template:Math. Given side lengths Template:Mvar we have:

Name; Symbol		Trilinear coordinates	Description
Vertices	Template:Math	$1:0:0$	Points at the corners of the triangle
	Template:Math	$0:1:0$
	Template:Math	$0:0:1$
Incenter	Template:Math	$1:1:1$	Intersection of the internal angle bisectors; Center of the triangle's inscribed circle
Excenters	Template:Math	$-1:1:1$	Intersections of the angle bisectors (two external, one internal); Centers of the triangle's three escribed circles
	Template:Math	$1:-1:1$
	Template:Math	$1:1:-1$
Centroid	Template:Math	$\frac{1}{a}:\frac{1}{b}:\frac{1}{c}$	Intersection of the medians; Center of mass of a uniform triangular lamina
Circumcenter	Template:Math	$\cos A:\cos B:\cos C$	Intersection of the perpendicular bisectors of the sides; Center of the triangle's circumscribed circle
Orthocenter	Template:Math	$\sec A:\sec B:\sec C$	Intersection of the altitudes
Nine-point center	Template:Math	$\begin{array}{cc}& \cos (B-C):\cos (C-A)\\ & :\cos (A-B)\end{array}$	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	Template:Math	$a:b:c$	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 Template:Math (these being proportional to actual signed areas of the triangles Template:Math, where Template:Mvar = centroid.)

The midpoint of, for example, side Template:Mvar has trilinear coordinates in actual sideline distances $(0,\frac{\mathrm{\Delta }}{b},\frac{\mathrm{\Delta }}{c})$ for triangle area Template:Math, which in arbitrarily specified relative distances simplifies to Template:Math. The coordinates in actual sideline distances of the foot of the altitude from Template:Mvar to Template:Mvar are $(0,\frac{2\mathrm{\Delta }}{a}\cos C,\frac{2\mathrm{\Delta }}{a}\cos B),$ which in purely relative distances simplifies to Template:Math.Template:Rp

Formulas

Collinearities and concurrencies

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

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

are collinear if and only if the determinant

$D=\left|\begin{array}{ccc}p& q& r\\ u& v& w\\ x& y& z\end{array}\right|$

equals zero. Thus if Template:Math is a variable point, the equation of a line through the points Template:Mvar and Template:Mvar is Template:Math.Template:Rp From this, every straight line has a linear equation homogeneous in Template:Mvar. Every equation of the form $lx+my+nz=0$ in real coefficients is a real straight line of finite points unless Template:Math is proportional to Template:Math, the side lengths, in which case we have the locus of points at infinity.Template:Rp

The dual of this proposition is that the lines

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

concur in a point Template:Math if and only if Template:Math.Template:Rp

Also, if the actual directed distances are used when evaluating the determinant of Template:Mvar, then the area of triangle Template:Math is Template:Mvar, where $K=\frac{-abc}{8{\mathrm{\Delta }}^2}$ (and where Template:Math is the area of triangle Template:Math, as above) if triangle Template:Math has the same orientation (clockwise or counterclockwise) as Template:Math, and $K=\frac{-abc}{8{\mathrm{\Delta }}^2}$ otherwise.

Parallel lines

Two lines with trilinear equations $lx+my+nz=0$ and $l^{\prime }x+m^{\prime }y+n^{\prime }z=0$ are parallel if and only ifTemplate:Rp

$\left|\begin{array}{ccc}l& m& n\\ l^{\prime }& m^{\prime }& n^{\prime }\\ a& b& c\end{array}\right|=0,$

where Template:Mvar are the side lengths.

Angle between two lines

The tangents of the angles between two lines with trilinear equations $lx+my+nz=0$ and $l^{\prime }x+m^{\prime }y+n^{\prime }z=0$ are given byTemplate:Rp

$\pm \frac{(m{n}^{\prime }-m^{\prime }n)\sin A+(n{l}^{\prime }-n^{\prime }l)\sin B+(l{m}^{\prime }-l^{\prime }m)\sin C}{l{l}^{\prime }+m{m}^{\prime }+n{n}^{\prime }-(m{n}^{\prime }+m^{\prime }n)\cos A-(n{l}^{\prime }+n^{\prime }l)\cos B-(l{m}^{\prime }+l^{\prime }m)\cos C}.$

Thus they are perpendicular ifTemplate:Rp

$l{l}^{\prime }+m{m}^{\prime }+n{n}^{\prime }-(m{n}^{\prime }+m^{\prime }n)\cos A-(n{l}^{\prime }+n^{\prime }l)\cos B-(l{m}^{\prime }+l^{\prime }m)\cos C=0.$

Altitude

The equation of the altitude from vertex Template:Mvar to side Template:Mvar isTemplate:Rp

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

Line in terms of distances from vertices

The equation of a line with variable distances Template:Mvar from the vertices Template:Mvar whose opposite sides are Template:Mvar isTemplate:Rp

$apx+bqy+crz=0.$

Actual-distance trilinear coordinates

The trilinears with the coordinate values Template:Mvar being the actual perpendicular distances to the sides satisfyTemplate:Rp

$a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime }=2\mathrm{\Delta }$

for triangle sides Template:Mvar and area Template:Math. This can be seen in the figure at the top of this article, with interior point Template:Mvar partitioning triangle Template:Math into three triangles Template:Math with respective areas $\frac{1}{2}a{a}^{\prime },\frac{1}{2}b{b}^{\prime },\frac{1}{2}c{c}^{\prime }.$

Distance between two points

The distance Template:Mvar between two points with actual-distance trilinears Template:Math is given byTemplate:Rp

$d^2{\sin }^{2}C=(a_1-a_2{)}^2+(b_1-b_2{)}^2+2(a_1-a_2)(b_1-b_2)\cos C$

or in a more symmetric way

$d^2=\frac{abc}{4{\mathrm{\Delta }}^2}(a(b_1-b_2)(c_2-c_1)+b(c_1-c_2)(a_2-a_1)+c(a_1-a_2)(b_2-b_1)).$

Distance from a point to a line

The distance Template:Mvar from a point Template:Math, in trilinear coordinates of actual distances, to a straight line $lx+my+nz=0$ isTemplate:Rp

$d=\frac{l{a}^{\prime }+m{b}^{\prime }+n{c}^{\prime }}{\sqrt{l^2+m^2+n^2-2mn\cos A-2nl\cos B-2lm\cos C}}.$

Quadratic curves

The equation of a conic section in the variable trilinear point Template:Math isTemplate:Rp

$r{x}^2+s{y}^2+t{z}^2+2uyz+2vzx+2wxy=0.$

It has no linear terms and no constant term.

The equation of a circle of radius Template:Mvar having center at actual-distance coordinates Template:Math isTemplate:Rp

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

Circumconics

The equation in trilinear coordinates Template:Mvar of any circumconic of a triangle isTemplate:Rp

$lyz+mzx+nxy=0.$

If the parameters Template:Mvar respectively equal the side lengths Template:Mvar (or the sines of the angles opposite them) then the equation gives the circumcircle.Template:Rp

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center Template:Math isTemplate:Rp

$yz(x^{\prime }-y^{\prime }-z^{\prime })+zx(y^{\prime }-z^{\prime }-x^{\prime })+xy(z^{\prime }-x^{\prime }-y^{\prime })=0.$

Inconics

Every conic section inscribed in a triangle has an equation in trilinear coordinates:Template:Rp

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

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

The equation of the incircle can be simplified toTemplate:Rp

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

while the equation for, for example, the excircle adjacent to the side segment opposite vertex Template:Mvar can be written asTemplate:Rp

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

Cubic curves

Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic Template:Math, as the locus of a point Template:Mvar such that the Template:Mvar-isoconjugate of Template:Mvar is on the line Template:Mvar is given by the determinant equation

$\left|\begin{array}{ccc}x& y& z\\ qryz& rpzx& pqxy\\ u& v& w\end{array}\right|=0.$

Among named cubics Template:Math are the following:

Thomson cubic: Template:Tmath, where Template:Tmath is centroid and Template:Tmath is incenter

Feuerbach cubic: Template:Tmath, where Template:Tmath is Feuerbach point

Darboux cubic: Template:Tmath, where Template:Tmath is De Longchamps point

Neuberg cubic: Template:Tmath, where Template:Tmath is Euler infinity point.

Conversions

Between trilinear coordinates and distances from sidelines

For any choice of trilinear coordinates Template:Math to locate a point, the actual distances of the point from the sidelines are given by Template:Math where Template:Mvar can be determined by the formula $k=\frac{2\mathrm{\Delta }}{ax+by+cz}$ in which Template:Mvar are the respective sidelengths Template:Mvar, and Template:Math is the area of Template:Math.

Between barycentric and trilinear coordinates

A point with trilinear coordinates Template:Math has barycentric coordinates Template:Math where Template:Mvar are the sidelengths of the triangle. Conversely, a point with barycentrics Template:Math has trilinear coordinates $\frac{\alpha }{a}:\frac{\beta }{b}:\frac{\gamma }{c}.$

Between Cartesian and trilinear coordinates

Given a reference triangle Template:Math, express the position of the vertex Template:Mvar in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector Template:Tmath using vertex Template:Mvar as the origin. Similarly define the position vector of vertex Template:Mvar as Template:Tmath Then any point Template:Mvar associated with the reference triangle Template:Math can be defined in a Cartesian system as a vector $\overrightarrow{P}=k_1\overrightarrow{A}+k_2\overrightarrow{B}.$ If this point Template:Mvar has trilinear coordinates Template:Math then the conversion formula from the coefficients Template:Math and Template:Math in the Cartesian representation to the trilinear coordinates is, for side lengths Template:Mvar opposite vertices Template:Mvar,

$x:y:z=\frac{k_1}{a}:\frac{k_2}{b}:\frac{1-k_1-k_2}{c},$

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

$k_1=\frac{ax}{ax+by+cz},k_2=\frac{by}{ax+by+cz}.$

More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors Template:Tmath and if the point Template:Mvar has trilinear coordinates Template:Math, then the Cartesian coordinates of Template:Tmath are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates Template:Mvar as the weights. Hence the conversion formula from the trilinear coordinates Template:Mvar to the vector of Cartesian coordinates Template:Tmath of the point is given by

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

where the side lengths are

$\begin{array}{cc}& |\overrightarrow{C}-\overrightarrow{B}|=a,\\ & |\overrightarrow{A}-\overrightarrow{C}|=b,\\ & |\overrightarrow{B}-\overrightarrow{A}|=c.\end{array}$

See also

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

External links

• Template:MathWorld
• Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for 64000 triangle centers.