Trilinear coordinates

Template:Short description Template:More sources needed

In geometry, the trilinear coordinates $x : y : z$ Script error: No such module "Check for unknown parameters". 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 $x : y$ Script error: No such module "Check for unknown parameters". 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 $y : z$ Script error: No such module "Check for unknown parameters". is the ratio of the perpendicular distances from the point to the sidelines opposite vertices Template:Mvar and Template:Mvar respectively; and likewise for $z : x$ Script error: No such module "Check for unknown parameters". 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, $ka' : kb' : kc'$ Script error: No such module "Check for unknown parameters". 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 $(a^{'}, b^{'}, c^{'})$ 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 $△ ABC$ Script error: No such module "Check for unknown parameters". are $1 : 1 : 1$ Script error: No such module "Check for unknown parameters".; that is, the (directed) distances from the incenter to the sidelines Template:Mvar are proportional to the actual distances denoted by $(r, r, r)$ Script error: No such module "Check for unknown parameters"., where Template:Mvar is the inradius of $△ ABC$ Script error: No such module "Check for unknown parameters".. Given side lengths Template:Mvar we have:

Name; Symbol		Trilinear coordinates	Description
Vertices	$A$ Script error: No such module "Check for unknown parameters".	$1 : 0 : 0$	Points at the corners of the triangle
	$B$ Script error: No such module "Check for unknown parameters".	$0 : 1 : 0$
	$C$ Script error: No such module "Check for unknown parameters".	$0 : 0 : 1$
Incenter	$I$ Script error: No such module "Check for unknown parameters".	$1 : 1 : 1$	Intersection of the internal angle bisectors; Center of the triangle's inscribed circle
Excenters	$I A$ Script error: No such module "Check for unknown parameters".	$- 1 : 1 : 1$	Intersections of the angle bisectors (two external, one internal); Centers of the triangle's three escribed circles
	$I B$ Script error: No such module "Check for unknown parameters".	$1 : - 1 : 1$
	$I C$ Script error: No such module "Check for unknown parameters".	$1 : 1 : - 1$
Centroid	$G$ Script error: No such module "Check for unknown parameters".	$\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$	Intersection of the medians; Center of mass of a uniform triangular lamina
Circumcenter	$O$ Script error: No such module "Check for unknown parameters".	$\cos A : \cos B : \cos C$	Intersection of the perpendicular bisectors of the sides; Center of the triangle's circumscribed circle
Orthocenter	$H$ Script error: No such module "Check for unknown parameters".	$\sec A : \sec B : \sec C$	Intersection of the altitudes
Nine-point center	$N$ Script error: No such module "Check for unknown parameters".	$\begin{aligned} \cos (B - C) : \cos (C - A) \\ : \cos (A - B) \end{aligned}$	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	$K$ Script error: No such module "Check for unknown parameters".	$a : b : c$	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 $1 : 1 : 1$ Script error: No such module "Check for unknown parameters". (these being proportional to actual signed areas of the triangles $△ BGC, △ CGA, △ AGB$ Script error: No such module "Check for unknown parameters"., where Template:Mvar = centroid.)

The midpoint of, for example, side Template:Mvar has trilinear coordinates in actual sideline distances $(0, \frac{Δ}{b}, \frac{Δ}{c})$ for triangle area $Δ$ Script error: No such module "Check for unknown parameters"., which in arbitrarily specified relative distances simplifies to $0 : ca : ab$ Script error: No such module "Check for unknown parameters".. The coordinates in actual sideline distances of the foot of the altitude from Template:Mvar to Template:Mvar are $(0, \frac{2 Δ}{a} \cos C, \frac{2 Δ}{a} \cos B),$ which in purely relative distances simplifies to $0 : cos C : cos B$ Script error: No such module "Check for unknown parameters"..^[1]Template:Rp

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{matrix} p & q & r \\ u & v & w \\ x & y & z \end{matrix} |

equals zero. Thus if $x : y : z$ Script error: No such module "Check for unknown parameters". is a variable point, the equation of a line through the points Template:Mvar and Template:Mvar is $D = 0$ Script error: No such module "Check for unknown parameters"..^[1]Template:Rp From this, every straight line has a linear equation homogeneous in Template:Mvar. Every equation of the form $l x + m y + n z = 0$ in real coefficients is a real straight line of finite points unless $l : m : n$ Script error: No such module "Check for unknown parameters". is proportional to $a : b : c$ Script error: No such module "Check for unknown parameters"., the side lengths, in which case we have the locus of points at infinity.^[1]Template:Rp

The dual of this proposition is that the lines

\begin{aligned} p α + q β + r γ & = 0 \\ u α + v β + w γ & = 0 \\ x α + y β + z γ & = 0 \end{aligned}

concur in a point $(α, β, γ)$ Script error: No such module "Check for unknown parameters". if and only if $D = 0$ Script error: No such module "Check for unknown parameters"..^[1]Template:Rp

Also, if the actual directed distances are used when evaluating the determinant of Template:Mvar, then the area of triangle $△ PUX$ Script error: No such module "Check for unknown parameters". is Template:Mvar, where $K = \frac{- a b c}{8 Δ^{2}}$ (and where $Δ$ Script error: No such module "Check for unknown parameters". is the area of triangle $△ ABC$ Script error: No such module "Check for unknown parameters"., as above) if triangle $△ PUX$ Script error: No such module "Check for unknown parameters". has the same orientation (clockwise or counterclockwise) as $△ ABC$ Script error: No such module "Check for unknown parameters"., and $K = \frac{- a b c}{8 Δ^{2}}$ otherwise.

Parallel lines

Two lines with trilinear equations $l x + m y + n z = 0$ and $l^{'} x + m^{'} y + n^{'} z = 0$ are parallel if and only if^[1]Template:Rp

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

where Template:Mvar are the side lengths.

Angle between two lines

The tangents of the angles between two lines with trilinear equations $l x + m y + n z = 0$ and $l^{'} x + m^{'} y + n^{'} z = 0$ are given by^[1]Template:Rp

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

Thus they are perpendicular if^[1]Template:Rp

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

Altitude

The equation of the altitude from vertex Template:Mvar to side Template:Mvar is^[1]Template: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 is^[1]Template:Rp

a p x + b q y + c r z = 0 .

Actual-distance trilinear coordinates

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

a a^{'} + b b^{'} + c c^{'} = 2 Δ

for triangle sides Template:Mvar and area $Δ$ Script error: No such module "Check for unknown parameters".. This can be seen in the figure at the top of this article, with interior point Template:Mvar partitioning triangle $△ ABC$ Script error: No such module "Check for unknown parameters". into three triangles $△ PBC, △ PCA, △ PAB$ Script error: No such module "Check for unknown parameters". with respective areas $\frac{1}{2} a a^{'}, \frac{1}{2} b b^{'}, \frac{1}{2} c c^{'} .$

Distance between two points

The distance Template:Mvar between two points with actual-distance trilinears $a i : b i : c i$ Script error: No such module "Check for unknown parameters". is given by^[1]Template: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{a b c}{4 Δ^{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 $a' : b' : c'$ Script error: No such module "Check for unknown parameters"., in trilinear coordinates of actual distances, to a straight line $l x + m y + n z = 0$ is^[1]Template:Rp

d = \frac{l a^{'} + m b^{'} + n c^{'}}{\sqrt{l^{2} + m^{2} + n^{2} - 2 m n \cos A - 2 n l \cos B - 2 l m \cos C}} .

Quadratic curves

The equation of a conic section in the variable trilinear point $x : y : z$ Script error: No such module "Check for unknown parameters". is^[1]Template:Rp

r x^{2} + s y^{2} + t z^{2} + 2 u y z + 2 v z x + 2 w x y = 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 $(a', b', c')$ Script error: No such module "Check for unknown parameters". is^[1]Template:Rp

(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 Template:Mvar of any circumconic of a triangle is^[1]Template:Rp

l y z + m z x + n x y = 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.^[1]Template:Rp

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center $x' : y' : z'$ Script error: No such module "Check for unknown parameters". is^[1]Template:Rp

y z (x^{'} - y^{'} - z^{'}) + z x (y^{'} - z^{'} - x^{'}) + x y (z^{'} - x^{'} - y^{'}) = 0 .

Inconics

Every conic section inscribed in a triangle has an equation in trilinear coordinates:^[1]Template:Rp

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

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

The equation of the incircle can be simplified to^[1]Template: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 as^[1]Template: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 $Z (U, P)$ Script error: No such module "Check for unknown parameters"., 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

| \begin{matrix} x & y & z \\ q r y z & r p z x & p q x y \\ u & v & w \end{matrix} | = 0 .

Among named cubics $Z (U, P)$ Script error: No such module "Check for unknown parameters". 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 $x : y : z$ Script error: No such module "Check for unknown parameters". to locate a point, the actual distances of the point from the sidelines are given by $a' = kx, b' = ky, c' = kz$ Script error: No such module "Check for unknown parameters". where Template:Mvar can be determined by the formula $k = \frac{2 Δ}{a x + b y + c z}$ in which Template:Mvar are the respective sidelengths Template:Mvar, and $∆$ Script error: No such module "Check for unknown parameters". is the area of $△ ABC$ Script error: No such module "Check for unknown parameters"..

Between barycentric and trilinear coordinates

A point with trilinear coordinates $x : y : z$ Script error: No such module "Check for unknown parameters". has barycentric coordinates $ax : by : cz$ Script error: No such module "Check for unknown parameters". where Template:Mvar are the sidelengths of the triangle. Conversely, a point with barycentrics $α : β : γ$ Script error: No such module "Check for unknown parameters". has trilinear coordinates $\frac{α}{a} : \frac{β}{b} : \frac{γ}{c} .$

Between Cartesian and trilinear coordinates

Given a reference triangle $△ ABC$ Script error: No such module "Check for unknown parameters"., 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 $△ ABC$ Script error: No such module "Check for unknown parameters". can be defined in a Cartesian system as a vector $\vec{P} = k_{1} \vec{A} + k_{2} \vec{B} .$ If this point Template:Mvar has trilinear coordinates $x : y : z$ Script error: No such module "Check for unknown parameters". then the conversion formula from the coefficients $k 1$ Script error: No such module "Check for unknown parameters". and $k 2$ Script error: No such module "Check for unknown parameters". 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{a x}{a x + b y + c z}, k_{2} = \frac{b y}{a x + b y + c z} .

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 $x : y : z$ Script error: No such module "Check for unknown parameters"., 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

\vec{P} = \frac{a x}{a x + b y + c z} \vec{A} + \frac{b y}{a x + b y + c z} \vec{B} + \frac{c z}{a x + b y + c z} \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

↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r ^s ^t 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.

Script error: No such module "Check for unknown parameters".

External links

Script error: No such module "Template wrapper".
Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for 64000 triangle centers.

Script error: No such module "Authority control".

[WW-1] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r ^s ^t 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.

[1]

Trilinear coordinates

Contents

Notation

Examples

Formulas

Collinearities and concurrencies

Parallel lines

Angle between two lines

Altitude

Line in terms of distances from vertices

Actual-distance trilinear coordinates

Distance between two points

Distance from a point to a line

Quadratic curves

Circumconics

Inconics

Cubic curves

Conversions

Between trilinear coordinates and distances from sidelines

Between barycentric and trilinear coordinates

Between Cartesian and trilinear coordinates

See also

References

External links

Navigation menu

Trilinear coordinates

Notation

Examples

Formulas

Collinearities and concurrencies

Parallel lines

Angle between two lines

Altitude

Line in terms of distances from vertices

Actual-distance trilinear coordinates

Distance between two points

Distance from a point to a line

Quadratic curves

Circumconics

Inconics

Cubic curves

Conversions

Between trilinear coordinates and distances from sidelines

Between barycentric and trilinear coordinates

Between Cartesian and trilinear coordinates

See also

References

External links

Navigation menu

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