Given a Triangle , the trilinear coordinates of a point with respect to are an ordered Triple of numbers, each of which is Proportional to the directed distance from to one of the side lines. Trilinear coordinates are denoted or and also are known as Barycentric Coordinates, Homogeneous Coordinates, or ``trilinears.''

In trilinear coordinates, the three Vertices , , and are given by , , and
. Let the point in the above diagram have trilinear coordinates
and lie at distances
, , and from the sides , , and , respectively. Then the distances , ,
and can be found by writing for the Area of , and similarly for
and . We then have

(1) |

so

(2) |

Trilinear coordinates are unchanged when each is multiplied by any constant , so

(3) |

(4) |

(5) |

(6) |

Trilinear coordinates for some common Points are summarized in the following table, where , , and are the angles at the corresponding vertices and , , and are the opposite side lengths.

Point | Trilinear Center Function |

Centroid | , |

Circumcenter | |

de Longchamps Point | |

Equal Detour Point | |

Feuerbach Point | |

Incenter | 1 |

Isoperimetric Point | |

Lemoine Point | |

Nine-Point Center | |

Orthocenter | |

Vertex | |

Vertex | |

Vertex |

To convert trilinear coordinates to a vector position for a given triangle specified by the - and -coordinates of
its axes, pick two Unit Vectors along the sides. For instance, pick

(7) | |||

(8) |

where these are the Unit Vectors and . Assume the Triangle has been labeled such that is the lower rightmost Vertex and . Then the Vectors obtained by traveling and along the sides and then inward Perpendicular to them must meet

(9) |

(10) | |||

(11) |

gives

(12) | |||

(13) |

But and are Unit Vectors, so

(14) | |||

(15) |

And the Vector coordinates of the point are then

(16) |

**References**

Boyer, C. B. *History of Analytic Geometry.* New York: Yeshiva University, 1956.

Casey, J. ``The General Equation--Trilinear Co-Ordinates.'' Ch. 10 in
*A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing
an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl.* Dublin: Hodges, Figgis, & Co., pp. 333-348, 1893.

Coolidge, J. L. *A Treatise on Algebraic Plane Curves.* New York: Dover, pp. 67-71, 1959.

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, 1969.

Coxeter, H. S. M. ``Some Applications of Trilinear Coordinates.'' *Linear Algebra Appl.* **226-228**, 375-388, 1995.

Kimberling, C. ``Triangle Centers and Central Triangles.'' *Congr. Numer.* **129**, 1-295, 1998.

© 1996-9

1999-05-26