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:

Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates $A$ (these being proportional to actual signed areas of the triangles $B$, 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 $C$, which in arbitrarily specified relative distances simplifies to $I$. 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 $I_{A}$.

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 $I_{B}$ is a variable point, the equation of a line through the points $P$ and $U$ is $I_{C}$. 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 $G$ is proportional to $O$, the side lengths, in which case we have the locus of points at infinity.

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 $H$ if and only if $N$.

Also, if the actual directed distances are used when evaluating the determinant of $D$, then the area of triangle $K$ is $KD$, where $$K = \tfrac{-abc}{8\Delta^2}$$ (and where $1 : 1 : 1$ is the area of triangle $△BGC, △CGA, △AGB$, as above) if triangle $Δ$ has the same orientation (clockwise or counterclockwise) as $0 : ca : ab$, 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


 * $$ \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


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

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


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

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


 * $$apx+bqy+crz=0.$$

Actual-distance trilinear coordinates
The trilinears with the coordinate values $a', b', c'$ being the actual perpendicular distances to the sides satisfy


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

for triangle sides $a, b, c$ and area $0 : cos C : cos B$. This can be seen in the figure at the top of this article, with interior point $P$ partitioning triangle $x : y : z$ into three triangles $D = 0$ 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 $l : m : n$ is given by


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

or in a more symmetric way


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

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


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

Quadratic curves
The equation of a conic section in the variable trilinear point $(α, β, γ)$ 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 $D = 0$ is


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

Circumconics
The equation in trilinear coordinates $x, y, z$ of any circumconic of a triangle is


 * $$lyz+mzx+nxy=0.$$

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 $△PUX$ is


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

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


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

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

The equation of the incircle can be simplified to


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

while the equation for, for example, the excircle adjacent to the side segment opposite vertex $A$ can be written as


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

Cubic curves
Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic $Δ$, 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 $△ABC$ are the following:


 * Thomson cubic: $Z(X(2),X(1))$, where $X(2)$ is centroid and $X(1)$ is incenter
 * Feuerbach cubic: $Z(X(5),X(1))$, where $X(5)$ is Feuerbach point
 * Darboux cubic: $Z(X(20),X(1))$, where $X(20)$ is De Longchamps point
 * Neuberg cubic: $Z(X(30),X(1))$, where $X(30)$ is Euler infinity point.

Between trilinear coordinates and distances from sidelines
For any choice of trilinear coordinates $△PUX$ to locate a point, the actual distances of the point from the sidelines are given by $△ABC$ 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 $Δ$ is the area of $△ABC$.

Between barycentric and trilinear coordinates
A point with trilinear coordinates $△PBC, △PCA, △PAB$ has barycentric coordinates $ai : bi : ci$ where $a, b, c$ are the sidelengths of the triangle. Conversely, a point with barycentrics $a' : b' : c'$ has trilinear coordinates $$\tfrac{\alpha}{a} : \tfrac{\beta}{b} : \tfrac{\gamma}{c}.$$

Between Cartesian and trilinear coordinates
Given a reference triangle $x : y : z$, 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 $(a', b', c' )$ 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 $Z(U, P)$ and $Z(U, P)$ 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{align}

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