View source for Trilinear coordinates

{{short description|Coordinate system based on distances from the sidelines of a given triangle}}
{{more sources needed|date=August 2024}}
[[Image:Trilinear coordinates.svg|right|300px]]

In [[geometry]], the '''trilinear coordinates''' {{math|''x'' : ''y'' : ''z''}} of a point relative to a given [[triangle]] describe the relative [[directed distance]]s from the three [[extended side|sidelines]] of the triangle.  Trilinear coordinates are an example of [[homogeneous coordinates]].  The ratio {{math|''x'' : ''y''}} is the ratio of the [[perpendicular distance]]s from the point to the sides (extended if necessary) opposite [[vertex (geometry)|vertices]] {{mvar|A}} and {{mvar|B}} respectively; the ratio {{math|''y'' : ''z''}} is the ratio of the perpendicular distances from the point to the sidelines opposite vertices {{mvar|B}} and {{mvar|C}} respectively; and likewise for {{math|''z'' : ''x''}} and vertices {{mvar|C}} and {{mvar|A}}.

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances ({{mvar|a'}}, {{mvar|b'}}, {{mvar|c'}}), or equivalently in ratio form, {{math|''ka' '': ''kb' '': ''kc' ''}} for any positive constant {{mvar|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 <math>x : y : z</math> for trilinear coordinates is often used in preference to the ordered [[tuple|triple]] notation <math>(x, y, z),</math> with the latter reserved for triples of directed distances <math>(a', b', c')</math> relative to a specific triangle. The trilinear coordinates <math>x : y : z,</math> can be rescaled by any arbitrary value without affecting their ratio. The bracketed, comma-separated triple notation <math>(x, y, z)</math> can cause confusion because conventionally this represents a different triple than e.g. <math>(2x, 2y, 2z),</math> but these equivalent ratios <math>x : y : z = {}\!</math><math>2x : 2y : 2z</math> represent the same point.

==Examples==

The trilinear coordinates of the [[incenter]] of a triangle {{math|△''ABC''}} are {{math|1 : 1 : 1}}; that is, the (directed) distances from the incenter to the sidelines {{mvar|BC, CA, AB}} are proportional to the actual distances denoted by {{math|(''r'', ''r'', ''r'')}}, where {{mvar|r}} is the inradius of {{math|△''ABC''}}. Given side lengths {{mvar|a, b, c}} we have:

{| class="wikitable" border="1" style="max-width:50em"
! colspan=2 | Name; Symbol
! Trilinear coordinates
! style="max-width:15em" |Description
|-
|rowspan=3 | Vertices
| style="text-align:center;" | {{math|''A''}}
| style="text-align:center;" | <math>1 : 0 : 0</math>
| rowspan=3 | Points at the corners of the triangle
|-
| style="text-align:center;" | {{math|''B''}}
| style="text-align:center;" | <math>0 : 1 : 0</math>
|-
| style="text-align:center;" | {{math|''C''}}
| style="text-align:center;" | <math>0 : 0 : 1</math>
|-
| [[Incenter]]
| style="text-align:center;" | {{math|''I''}}
| style="text-align:center;" | <math>1 : 1 : 1</math>
| Intersection of the internal [[angle bisectors]];  Center of the triangle's [[incircle|inscribed circle]]
|-
| rowspan=3 |[[Excenter]]s
| style="text-align:center;" | {{math|''I''<sub>''A''</sub>}}
| style="text-align:center;" | <math>-1 : 1 : 1</math>
| rowspan=3| Intersections of the angle bisectors (two external, one internal); Centers of the triangle's three [[excircle|escribed circles]]
|-
| style="text-align:center;" | {{math|''I''<sub>''B''</sub>}}
| style="text-align:center;" | <math>1 : -1 : 1</math>
|-
| style="text-align:center;" | {{math|''I''<sub>''C''</sub>}}
| style="text-align:center;" | <math>1 : 1 : -1</math>
|-
| [[Centroid#Of triangle and tetrahedron|Centroid]]
| style="text-align:center;" | {{math|''G''}}
| style="text-align:center;" | <math>\frac1a : \frac1b : \frac1c</math>
| Intersection of the [[median (geometry)|medians]];  [[Center of mass]] of a uniform triangular [[planar lamina|lamina]]
|-
| [[Circumcenter]]
| style="text-align:center;" | {{math|''O''}}
| style="text-align:center;" | <math>\cos A : \cos B : \cos C</math>
| Intersection of the [[perpendicular bisector]]s of the sides; Center of the triangle's [[circumcircle|circumscribed circle]]
|-
| [[Orthocenter]]
| style="text-align:center;" | {{math|''H''}}
| style="text-align:center;" | <math>\sec A : \sec B : \sec C</math>
| Intersection of the [[altitude (triangle)|altitudes]]
|-
| [[Nine-point center]]
| style="text-align:center;" | {{math|''N''}}
| style="text-align:center;" | <math>\begin{align}&\cos(B-C) : \cos(C-A) \\ &\qquad : \cos(A-B)\end{align}</math>
| 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]]
| style="text-align:center;" | {{math|''K''}}
| style="text-align:center;" | <math>a : b : c</math>
| 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 (mathematics)|barycentric coordinates]] {{math|1 : 1 : 1}} (these being proportional to actual signed areas of the triangles {{math|△''BGC'', △''CGA'', △''AGB''}}, where {{mvar|G}} = centroid.)

The midpoint of, for example, side {{mvar|BC}} has trilinear coordinates in actual sideline distances  <math>(0 , \tfrac{\Delta}{b} , \tfrac{\Delta}{c}) </math> for triangle area {{math|Δ}}, which in arbitrarily specified relative distances simplifies to {{math|0 : ''ca'' : ''ab''}}. The coordinates in actual sideline distances of the foot of the altitude from {{mvar|A}} to {{mvar|BC}} are <math>(0, \tfrac{2\Delta}{a}\cos C, \tfrac{2\Delta}{a}\cos B),</math> which in purely relative distances simplifies to {{math|0 : cos ''C'' : cos ''B''}}.<ref name=WW/>{{rp|p. 96}}

==Formulas==

===Collinearities and concurrencies===

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

:<math>\begin{align}
P &= p:q:r \\
U &= u:v:w \\
X &= x:y:z \\
\end{align}</math>

are [[collinearity|collinear]] if and only if the [[determinant]]

:<math> D = \begin{vmatrix}
p & q & r \\
u & v & w \\
x & y & z
\end{vmatrix}</math>

equals zero.  Thus if {{math|''x'' : ''y'' : ''z''}} is a variable point, the equation of a line through the points {{mvar|P}} and {{mvar|U}} is {{math|1=''D'' = 0}}.<ref name=WW>William Allen Whitworth (1866) [http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01190002 Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise], link from [[Cornell University]] Historical Math Monographs.</ref>{{rp|p. 23}} From this, every straight line has a linear equation homogeneous in {{mvar|x, y, z}}.  Every equation of the form <math>lx+my+nz=0</math> in real coefficients is a real straight line of finite points unless {{math|''l'' : ''m'' : ''n''}} is proportional to {{math|''a'' : ''b'' : ''c''}}, the side lengths, in which case we have the locus of points at infinity.<ref name=WW/>{{rp|p. 40}}

The dual of this proposition is that the lines

:<math>\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}</math>

[[concurrent lines|concur]] in a point {{math|(''α'', ''β'', ''γ'')}} if and only if {{math|1=''D'' = 0}}.<ref name="WW"/>{{rp|p. 28}}

Also, if the actual directed distances are used when evaluating the determinant of {{mvar|D}}, then the area of triangle {{math|△''PUX''}} is {{mvar|KD}}, where <math>K = \tfrac{-abc}{8\Delta^2}</math> (and where {{math|Δ}} is the area of triangle {{math|△''ABC''}}, as above) if triangle {{math|△''PUX''}} has the same orientation (clockwise or counterclockwise) as {{math|△''ABC''}}, and <math>K = \tfrac{-abc}{8\Delta^2}</math> otherwise.

===Parallel lines===

Two lines with trilinear equations <math>lx+my+nz=0</math> and <math>l'x+m'y+n'z=0</math> are parallel if and only if<ref name=WW/>{{rp|p. 98,#xi}}

:<math> \begin{vmatrix}
l  & m  & n \\
l' & m' & n' \\
a  & b  & c 
\end{vmatrix}=0,</math>

where {{mvar|a, b, c}} are the side lengths.

===Angle between two lines===

The [[tangent (trigonometry)|tangents]] of the angles between two lines with trilinear equations <math>lx+my+nz=0</math> and <math>l'x+m'y+n'z=0</math> are given by<ref name=WW/>{{rp|at=Art. 48}}

:<math>\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}.</math>

Thus they are perpendicular if<ref name=WW/>{{rp|at=Art. 49}}

:<math>ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0.</math>

===Altitude===

The equation of the [[altitude (geometry)|altitude]] from vertex {{mvar|A}} to side {{mvar|BC}} is<ref name=WW/>{{rp|p.98,#x}}

:<math>y\cos B-z\cos C=0.</math>

===Line in terms of distances from vertices===

The equation of a line with variable distances {{mvar|p, q, r}} from the vertices {{mvar|A, B, C}} whose opposite sides are {{mvar|a, b, c}} is<ref name=WW/>{{rp|p. 97,#viii}}

:<math>apx+bqy+crz=0.</math>

===Actual-distance trilinear coordinates===

The trilinears with the coordinate values {{mvar|a', b', c'}}  being the actual perpendicular distances to the sides satisfy<ref name=WW/>{{rp|p. 11}}

:<math>aa' +bb' + cc' =2\Delta</math>

for triangle sides {{mvar|a, b, c}} and area {{math|Δ}}. This can be seen in the figure at the top of this article, with interior point {{mvar|P}} partitioning triangle {{math|△''ABC''}} into three triangles {{math|△''PBC'', △''PCA'', △''PAB''}} with respective areas <math>\tfrac{1}{2}aa' , \tfrac{1}{2}bb', \tfrac{1}{2}cc'.</math>

===Distance between two points===

The distance {{mvar|d}} between two points with actual-distance trilinears {{math|''a{{sub|i}}'' : ''b{{sub|i}}'' : ''c{{sub|i}}''}} is given by<ref name=WW/>{{rp|p. 46}}

:<math>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</math>

or in a more symmetric way

:<math>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).</math>

===Distance from a point to a line===

The distance {{mvar|d}} from a point {{math|''a' '': ''b' '': ''c' ''}}, in trilinear coordinates of actual distances, to a straight line <math>lx+my+nz=0</math> is<ref name=WW/>{{rp|p. 48}}

:<math>d=\frac{la'+mb'+nc'}{\sqrt{l^2+m^2+n^2-2mn\cos A -2nl\cos B -2lm\cos C}}.</math>

===Quadratic curves===

The equation of a [[conic section]] in the variable trilinear point {{math|''x'' : ''y'' : ''z''}} is<ref name=WW/>{{rp|p.118}}

:<math>rx^2+sy^2+tz^2+2uyz+2vzx+2wxy=0.</math>

It has no linear terms and no constant term.

The equation of a circle of radius {{mvar|r}} having center at actual-distance coordinates {{math|(''a', b', c' '')}} is<ref name=WW/>{{rp|p.287}}

:<math>(x-a')^2\sin 2A+(y-b')^2\sin 2B+(z-c')^2\sin 2C=2r^2\sin A\sin B\sin C.</math>

====Circumconics====

The equation in trilinear coordinates {{mvar|x, y, z}} of any [[circumconic]] of a triangle is<ref name=WW/>{{rp|p. 192}}

:<math>lyz+mzx+nxy=0.</math>

If the parameters {{mvar|l, m, n}} respectively equal the side lengths {{mvar|a, b, c}} (or the sines of the angles opposite them) then the equation gives the [[circumcircle]].<ref name=WW/>{{rp|p. 199}}

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center {{math|''x' '':  ''y' '': ''z' ''}} is<ref name=WW/>{{rp|p. 203}}

:<math>yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0.</math>

====Inconics====

Every conic section [[inscribed figure|inscribed]] in a triangle has an equation in trilinear coordinates:<ref name=WW/>{{rp|p. 208}}

:<math>l^2x^2+m^2y^2+n^2z^2 \pm 2mnyz \pm 2nlzx\pm 2lmxy =0,</math>

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

The equation of the [[incircle]] can be simplified to<ref name=WW/>{{rp|p. 210, p.214}}

:<math>\pm \sqrt{x}\cos \frac{A}{2}\pm \sqrt{y}\cos \frac{B}{2}\pm\sqrt{z}\cos \frac{C}{2}=0,</math>

while the equation for, for example, the [[excircle]] adjacent to the side segment opposite vertex {{mvar|A}} can be written as<ref name= WW/>{{rp|p. 215}}

:<math>\pm \sqrt{-x}\cos \frac{A}{2}\pm \sqrt{y}\cos \frac{B}{2}\pm\sqrt{z}\cos \frac{C}{2}=0.</math>

===Cubic curves===
Many cubic curves are easily represented using trilinear coordinates.  For example, the pivotal self-isoconjugate cubic {{math|''Z''(''U, P'')}}, as the locus of a point {{mvar|X}} such that the {{mvar|P}}-isoconjugate of {{mvar|X}} is on the line {{mvar|UX}} is given by the determinant equation

:<math> \begin{vmatrix}x&y&z\\
qryz&rpzx&pqxy\\
u&v&w\end{vmatrix} = 0.</math>

Among named cubics {{math|''Z''(''U, P'')}} are the following:

: [[Thomson cubic]]: {{tmath|Z(X(2),X(1))}}, where {{tmath|X(2)}} is ''[[centroid]] and {{tmath|X(1)}} is ''[[incenter]]
: [[Cubic plane curve#Napoleon–Feuerbach cubic|Feuerbach cubic]]: {{tmath|Z(X(5),X(1))}}, where {{tmath|X(5)}} is ''[[Feuerbach point]]''
: [[Cubic plane curve#Darboux cubic|Darboux cubic]]: {{tmath|Z(X(20),X(1))}}, where {{tmath|X(20)}} is ''[[De Longchamps point]]''
: [[Neuberg cubic]]: {{tmath|Z(X(30),X(1))}}, where {{tmath|X(30)}} is ''[[Euler infinity point]]''.

==Conversions==

===Between trilinear coordinates and distances from sidelines===

For any choice of trilinear coordinates {{math|''x'' : ''y'' : ''z''}} to locate a point, the actual distances of the point from the sidelines are given by {{math|1=''a' ''= ''kx'', ''b' ''= ''ky'', ''c' ''= ''kz''}} where {{mvar|k}} can be determined by the formula <math>k = \tfrac{2\Delta}{ax + by + cz}</math> in which {{mvar|a, b, c}} are the respective sidelengths {{mvar|BC, CA, AB}}, and {{math|∆}} is the area of {{math|△''ABC''}}.

===Between barycentric and trilinear coordinates===
A point with trilinear coordinates {{math|''x'' : ''y'' : ''z''}} has [[barycentric coordinates (mathematics)|barycentric coordinates]] {{math|''ax'' : ''by'' : ''cz''}} where {{mvar|a, b, c}} are the sidelengths of the triangle.  Conversely, a point with barycentrics {{math|''α'' : ''β'' : ''γ''}} has trilinear coordinates <math>\tfrac{\alpha}{a} : \tfrac{\beta}{b} : \tfrac{\gamma}{c}.</math>

===Between Cartesian and trilinear coordinates===
Given a reference triangle {{math|△''ABC''}}, express the position of the vertex {{mvar|B}} in terms of an ordered pair of [[Cartesian coordinates]] and represent this algebraically as a [[vector (geometric)|vector]] {{tmath|\vec B,}} using vertex {{mvar|C}} as the origin. Similarly define the position vector of vertex {{mvar|A}} as {{tmath|\vec A.}} Then any point {{mvar|P}} associated with the reference triangle {{math|△''ABC''}} can be defined in a Cartesian system as a vector <math>\vec P = k_1\vec A + k_2\vec B.</math> If this point {{mvar|P}} has trilinear coordinates {{math|''x'' : ''y'' : ''z''}} then the conversion formula from the coefficients {{math|''k''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}} in the Cartesian representation to the trilinear coordinates is, for side lengths {{mvar|a, b, c}} opposite vertices {{mvar|A, B, C}},

: <math>x:y:z = \frac{k_1}{a} : \frac{k_2}{b} : \frac{1 - k_1 - k_2}{c}, </math>

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

: <math>k_1 = \frac{ax}{ax + by + cz}, \quad k_2 = \frac{by}{ax + by + cz}.</math>

More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors {{tmath|\vec A, \vec B, \vec C}} and if the point {{mvar|P}} has trilinear coordinates {{math|''x'' : ''y'' : ''z''}}, then the Cartesian coordinates of {{tmath|\vec P}} are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates {{mvar|ax, by, cz}} as the weights. Hence the conversion formula from the trilinear coordinates {{mvar|x, y, z}}  to the vector of Cartesian coordinates {{tmath|\vec P}} of the point is given by

: <math>\vec{P}=\frac{ax}{ax+by+cz}\vec{A}+\frac{by}{ax+by+cz}\vec{B}+\frac{cz}{ax+by+cz}\vec{C},</math>

where the side lengths are 
:<math>\begin{align}
& |\vec C - \vec B| = a, \\
& |\vec A - \vec C| = b, \\
& |\vec B - \vec A| = c.
\end{align}</math>

==See also==

*[[Morley's triangle]]s, giving examples of numerous points expressed in trilinear coordinates
*[[Ternary plot]]
*[[Viviani's theorem]]

==References==
{{Reflist}}

==External links==
*{{MathWorld|title=Trilinear Coordinates|urlname=TrilinearCoordinates}}
*[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers - ETC] by Clark Kimberling; has trilinear coordinates (and barycentric) for 64000 triangle centers.

{{Authority control}}

[[Category:Linear algebra]]
[[Category:Affine geometry]]
[[Category:Triangle geometry]]
[[Category:Coordinate systems]]