Trilinear coordinates explained

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

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances, or equivalently in ratio form, for any positive constant . 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. ( 2 x, 2 y, 2 z), but these equivalent ratios x : y : z = {} 2 x : 2 y : 2 z represent the same point.

Examples

The trilinear coordinates of the incenter of a triangle are ; that is, the (directed) distances from the incenter to the sidelines are proportional to the actual distances denoted by, where is the inradius of . Given side lengths we have:

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

The midpoint of, for example, side has trilinear coordinates in actual sideline distances

( 0 , \tfrac{\Delta}{b} , \tfrac{\Delta}{c}) for triangle area, which in arbitrarily specified relative distances simplifies to . The coordinates in actual sideline distances of the foot of the altitude from to are ( 0 , \tfrac{ 2 \Delta}{a}\cos C, \tfrac{ 2 \Delta}{a}\cos B), which in purely relative distances simplifies to .

Formulas

Collinearities and concurrencies

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

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

equals zero. Thus if is a variable point, the equation of a line through the points and is .^[1] From this, every straight line has a linear equation homogeneous in . Every equation of the form

lx + my + nz =0 in real coefficients is a real straight line of finite points unless is proportional to, the side lengths, in which case we have the locus of points at infinity.^[1]

The dual of this proposition is that the lines

\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 if and only if .^[1]

Also, if the actual directed distances are used when evaluating the determinant of, then the area of triangle is, where

K = \tfrac{ - abc}{ 8 \Delta ^{2}} (and where is the area of triangle, as above) if triangle has the same orientation (clockwise or counterclockwise) as, 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] \begin{vmatrix} l & m & n \\ l' & m' & n' \\ a & b & c \end{vmatrix} =0 ,

where 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] \pm ((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 ifll' + 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 to side is^[1]

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

Line in terms of distances from vertices

The equation of a line with variable distances from the vertices whose opposite sides are is^[1]

apx + bqy + crz =0 .

Actual-distance trilinear coordinates

The trilinears with the coordinate values being the actual perpendicular distances to the sides satisfy^[1]

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

for triangle sides and area . This can be seen in the figure at the top of this article, with interior point partitioning triangle into three triangles with respective areas

\tfrac{ 1 }{ 2 }aa' , \tfrac{ 1 }{ 2 }bb', \tfrac{ 1 }{ 2 }cc'.

Distance between two points

The distance between two points with actual-distance trilinears is given by^[1]

d ^{2\sin} ^{2} C = (a _{1-a} ^{2+ (b} ^{1- b} ^{2+2 (a} ^{1- a} _{2)(b} _{1-b} _{2)\cos} C

or in a more symmetric way

d ^{2} = (a b c)/(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 from a point, in trilinear coordinates of actual distances, to a straight line

lx + my + nz =0 is^[1] d = (la' + mb' + nc')/(\sqrt{l ^{2+m} ^{2+n} ^{2-2mn\cos} A -2 nl\cos B -2 lm\cos C) }.

Quadratic curves

The equation of a conic section in the variable trilinear point is^[1]

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 having center at actual-distance coordinates is^[1]

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

Circumconics

The equation in trilinear coordinates of any circumconic of a triangle is^[1]

lyz + mzx + nxy =0 .

If the parameters respectively equal the side lengths (or the sines of the angles opposite them) then the equation gives the circumcircle.^[1]

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center is^[1]

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]

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

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

The equation of the incircle can be simplified to^[1]

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

while the equation for, for example, the excircle adjacent to the side segment opposite vertex can be written as^[1]

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

Cubic curves

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

\begin{vmatrix}x&y&z\\ qryz&rpzx&pqxy\\ u&v&w\end{vmatrix} = 0 .

Among named cubics are the following:

Thomson cubic:, where is centroid and is incenter

Feuerbach cubic:, where is Feuerbach point

Darboux cubic:, where is De Longchamps point

Neuberg cubic:, where is Euler infinity point.

Conversions

Between trilinear coordinates and distances from sidelines

For any choice of trilinear coordinates to locate a point, the actual distances of the point from the sidelines are given by where can be determined by the formula

k = \tfrac{ 2 \Delta}{ax + by + cz} in which are the respective sidelengths, and is the area of .

Between barycentric and trilinear coordinates

A point with trilinear coordinates has barycentric coordinates where are the sidelengths of the triangle. Conversely, a point with barycentrics has trilinear coordinates

\tfrac{\alpha}{a} : \tfrac{\beta}{b} : \tfrac{\gamma}{c}.

Between Cartesian and trilinear coordinates

Given a reference triangle, express the position of the vertex in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector using vertex as the origin. Similarly define the position vector of vertex as Then any point associated with the reference triangle can be defined in a Cartesian system as a vector

\vec P = k _{1\vec} A + k _{2\vec} B. If this point has trilinear coordinates then the conversion formula from the coefficients and in the Cartesian representation to the trilinear coordinates is, for side lengths opposite vertices,x:y:z = (k _{1})/(a) : (k _{2})/(b) : (1- k _{1} - k _{2})/(c) ,

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

k _{1} = (ax)/(ax + by + cz) , k _{2} = (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 and if the point has trilinear coordinates, then the Cartesian coordinates of are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates as the weights. Hence the conversion formula from the trilinear coordinates to the vector of Cartesian coordinates of the point is given by

\vec{P} = (ax)/(\vec{A} + ax + by + cz) (by)/(\vec{B} + ax + by + cz) (cz)/(ax + by + cz) \vec{C},

where the side lengths are

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

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

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

Notes and References

1. 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.