300px
In

Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise

link from^{''2''} (and where ''∆'' is the area of triangle ''ABC'', as above) if triangle ''PUX'' has the same orientation (clockwise or counterclockwise) as triangle ''ABC'', and ''K'' = ''–abc/8∆''^{''2''} otherwise.

_{''i''} : ''b''_{''i''} : ''c''_{''i''} is given by
:$d^2\backslash sin\; ^2\; C=(a\_1-a\_2)^2+(b\_1-b\_2)^2+2(a\_1-a\_2)(b\_1-b\_2)\backslash cos\; C$
or in a more symmetric way
:$d^2\; =\; \backslash frac\backslash left(a(b\_1-b\_2)(c\_2-c\_1)+b(c\_1-c\_2)(a\_2-a\_1)+c(a\_1-a\_2)(b\_2-b\_1)\backslash right)$.

__''B''__, using vertex ''C'' as the origin. Similarly define the position vector of vertex ''A'' as __''A''__. Then any point ''P'' associated with the reference triangle ''ABC'' can be defined in a Cartesian system as a vector __''P''__ = ''k''_{1}__''A''__ + ''k''_{2}__''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'',
: $x:y:z\; =\; \backslash frac\; :\; \backslash frac\; :\; \backslash frac,$
and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is
: $k\_1\; =\; \backslash frac,\; \backslash quad\; k\_2\; =\; \backslash frac.$
More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors __''A''__, __''B''__ and __''C''__ and if the point ''P'' has trilinear coordinates ''x'' : ''y'' : ''z'', then the Cartesian coordinates of __''P''__ are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates ''ax'', ''by'' and ''cz'' as the weights. Hence the conversion formula from the trilinear coordinates ''x, y, z'' to the vector of Cartesian coordinates __''P''__ of the point is given by
: $\backslash underline=\backslash frac\backslash underline+\backslash frac\backslash underline+\backslash frac\backslash underline,$
where the side lengths are , __''C''__ − __''B''__, = ''a'', , __''A''__ − __''C''__, = ''b'' and , __''B''__ − __''A''__, = ''c''.

Encyclopedia of Triangle Centers - ETC

by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 7000 triangle centers {{Authority control Linear algebra Affine geometry Triangle geometry Coordinate systems

geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ...

, the trilinear coordinates ''x:y:z'' of a point relative to a given triangle
A triangle is a polygon
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is conce ...

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.
The name "trilinear coordinates" is sometimes abbreviated to "trilinears".
Notation

The ratio notation ''x'':''y'':''z'' for trilinear coordinates is different from the ordered triple notation (''a' '', ''b' '', ''c' '') for actual directed distances. Here each of ''x'', ''y'', and ''z'' has no meaning by itself; its ratio to one of the others ''does'' have meaning. Thus "comma notation" for trilinear coordinates should be avoided, because the notation (''x'', ''y'', ''z''), which means an ordered triple, does not allow, for example, (''x'', ''y'', ''z'') = (2''x'', 2''y'', 2''z''), whereas the "colon notation" does allow ''x'' : ''y'' : ''z'' = 2''x'' : 2''y'' : 2''z''.Examples

The trilinear coordinates of the 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 triangle ''ABC''. Given side lengths ''a, b, c'' we have: :* ''A'' = 1 : 0 : 0 :* ''B'' = 0 : 1 : 0 :* ''C'' = 0 : 0 : 1 :* = 1 : 1 : 1 :*centroid
In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a ...

= ''bc'' : ''ca'' : ''ab'' = 1/''a'' : 1/''b'' : 1/''c'' = csc ''A'' : csc ''B'' : csc ''C''.
:* circumcenter = cos ''A'' : cos ''B'' : cos ''C''.
:* orthocenter = sec ''A'' : sec ''B'' : sec ''C''.
:* nine-point center = cos(''B'' − ''C'') : cos(''C'' − ''A'') : cos(''A'' − ''B'').
:* symmedian point = ''a'' : ''b'' : ''c'' = sin ''A'' : sin ''B'' : sin ''C''.
:* ''A''-excenter = −1 : 1 : 1
:* ''B''-excenter = 1 : −1 : 1
:* ''C''-excenter = 1 : 1 : −1.
Note that, in general, the incenter is not the same as the centroid
In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a ...

; 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 $(0\; ,\; \backslash frac\; ,\; \backslash frac)$ for triangle area $\backslash Delta$, 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 $(0,\; \backslash frac\backslash cos\; C,\; \backslash frac\backslash cos\; B),$ which in purely relative distances simplifies to $0:\backslash cos\; C:\backslash cos\; B.$
Formulas

Collinearities and concurrencies

Trilinear coordinates enable many algebraic methods in triangle geometry. For example, three points :''P = p'' : ''q'' : ''r'' :''U = u'' : ''v'' : ''w'' :''X = x'' : ''y'' : ''z'' arecollinear
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

if and only if the determinant
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

:$D\; =\; \backslash beginp\&q\&r\backslash \backslash \; u\&v\&w\backslash \backslash x\&y\&z\backslash end$
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.William Allen Whitworth (1866Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise

link from

Cornell University
Cornell University ( ) is a private, statutory, Ivy League and land-grant research university in Ithaca, New York. Founded in 1865 by Ezra Cornell and Andrew Dickson White, the university was intended to teach and make contributions in a ...

Historical Math Monographs. 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.
The dual of this proposition is that the lines
:''pα + qβ + rγ = 0''
:''uα + vβ + wγ = 0'',
:''xα + yβ + zγ = 0''
concur in a point (α, β, γ) if and only if ''D'' = 0.
Also, if the actual directed distances are used when evaluating the determinant of ''D'', then the area of triangle ''PUX'' is ''KD'', where ''K'' = ''abc/8∆''Parallel lines

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 :$\backslash beginl\&m\&n\backslash \backslash \; l\text{'}\&m\text{'}\&n\text{'}\backslash \backslash a\&b\&c\backslash end=0,$ 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 $l\text{'}x+m\text{'}y+n\text{'}z=0$ are given by :$\backslash pm\; \backslash frac.$Perpendicular lines

Thus two lines with trilinear equations $lx+my+nz=0$ and $l\text{'}x+m\text{'}y+n\text{'}z=0$ are perpendicular if and only if :$ll\text{'}+mm\text{'}+nn\text{'}-(mn\text{'}+m\text{'}n)\backslash cos\; A-(nl\text{'}+n\text{'}l)\backslash cos\; B-(lm\text{'}+l\text{'}m)\backslash cos\; C=0.$Altitude

The equation of the altitude (geometry), altitude from vertex ''A'' to side ''BC'' is :$y\backslash cos\; B-z\backslash cos\; C=0.$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 :$apx+bqy+crz=0.$Actual-distance trilinear coordinates

The trilinears with the coordinate values ''a', b', c' '' being the actual perpendicular distances to the sides satisfy :$aa\text{'}\; +bb\text{'}\; +\; cc\text{'}\; =2\backslash Delta$ for triangle sides ''a, b, c'' and area $\backslash Delta$. 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'', and ''PAB'' with respective areas (1/2)''aa' '', (1/2)''bb' '', and (1/2)''cc' ''.Distance between two points

The distance ''d'' between two points with actual-distance trilinears ''a''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 :$d=\backslash frac.$Quadratic curves

The equation of a conic section in the variable trilinear point ''x'' : ''y'' : ''z'' is :$rx^2+sy^2+tz^2+2uyz+2vzx+2wxy=0.$ 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 :$(x-a\text{'})^2\backslash sin\; 2A+(y-b\text{'})^2\backslash sin\; 2B+(z-c\text{'})^2\backslash sin\; 2C=2r^2\backslash sin\; A\backslash sin\; B\backslash sin\; C.$Circumconics

The equation in trilinear coordinates ''x, y, z'' of any circumconic of a triangle is :$lyz+mzx+nxy=0.$ 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. Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center ''x' : y' : z' '' is :$yz(x\text{'}-y\text{'}-z\text{'})+zx(y\text{'}-z\text{'}-x\text{'})+xy(z\text{'}-x\text{'}-y\text{'})=0.$Inconics

Every conic section inscribed figure, inscribed in a triangle has an equation in trilinear coordinates: :$l^2x^2+m^2y^2+n^2z^2\; \backslash pm\; 2mnyz\; \backslash pm\; 2nlzx\backslash pm\; 2lmxy\; =0,$ with exactly one or three of the unspecified signs being negative. The equation of the incircle can be simplified to :$\backslash pm\; \backslash sqrt\backslash cos\; \backslash frac\backslash pm\; \backslash sqrt\backslash cos\; \backslash frac\backslash pm\backslash sqrt\backslash cos\; \backslash frac=0,$ while the equation for, for example, the excircle adjacent to the side segment opposite vertex ''A'' can be written as :$\backslash pm\; \backslash sqrt\backslash cos\; \backslash frac\backslash pm\; \backslash sqrt\backslash cos\; \backslash frac\backslash pm\backslash sqrt\backslash cos\; \backslash frac=0.$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 :$\backslash beginx\&y\&z\backslash \backslash \; qryz\&rpzx\&pqxy\backslash \backslash u\&v\&w\backslash end\; =\; 0.$ Among named cubics ''Z(U,P)'' are the following: : Thomson cubic: ''Z(X(2),X(1))'', where ''X(2) = ''centroid
In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a ...

, ''X(1) = ''
: Cubic plane curve#Napoleon–Feuerbach cubic, Feuerbach cubic: ''Z(X(5),X(1))'', where ''X(5) = ''Feuerbach point
: Cubic plane curve#Darboux cubic, Darboux cubic: ''Z(X(20),X(1))'', where ''X(20) = ''De Longchamps point
: Cubic plane curve#Neuberg cubic, 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 $k\; =\; \backslash frac$ 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 (mathematics), 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 ''α/a'' : ''β/b'' : ''γ/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 (geometric), vectorSee also

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

External links

*Encyclopedia of Triangle Centers - ETC

by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 7000 triangle centers {{Authority control Linear algebra Affine geometry Triangle geometry Coordinate systems